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Ordinary differential equations
Computational Neuroscience. Session 1-2
Dr. Marco A Roque Sol
05/29/2018
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation
is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives,
either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential Equations
A differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation
is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE,
if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation
is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation,
abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE,
if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Differential Equations
A differential equation is any equation which containsderivatives, either ordinary or partial derivatives of an unknownfunction.
Ordinary and Partial Differential EquationsA differential equation is called an ordinary differential equation,abbreviated by ODE, if it has only ordinary derivatives in it,having the form:
F (y (n), y (n−1), ..., y ′, y(t), t) = 0
Likewise, a differential equation is called a partial differentialequation, abbreviated by PDE, if it has partial derivatives in it.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics,
if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m
is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration a
and being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F
then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation.
First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a,
in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or
a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object
and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t .
We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force,
F may also be a function of time,velocity, and/or position.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
There many situations where we can find such an object. Thus,for instance in the case of the study of Classical Mechanics inPhysics, if an object of mass m is moving with acceleration aand being acted on with force F then NewtonâAZs Second Lawtells us.
F = ma
To see that this is in fact a differential equation. First, rememberthat we can rewrite the acceleration, a, in one of two ways.
a =dvdt
or a =d2udt2
Where v is the velocity of the object and u is the positionfunction of the object at any time t . We should also rememberat this point that the force, F may also be a function of time,velocity, and/or position.Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
mdvdt
= F (t , v) or md2udt2 = F (t , u, v)
More examples of differential equations
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
∂u3
∂2x∂t= 1 +
∂u∂y
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Order
The order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order
of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation
is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation.
The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
OrderThe order of a differential equation is the largest derivativepresent in the differential equation. The equation
mdvdt
= F (t , v)
is a first order differential equation, the equations
md2udt2 = F (t , u, v)
2y ′′ + 3y ′ − 5y = 0
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation
∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order
does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not
you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or
partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
a2 ∂2u∂x2 =
∂u∂t
a2 ∂2u∂x2 =
∂2u∂t2
are second order differential equations, the equation∂u3
∂2x∂t= 1 +
∂u∂y
is a third order differential equation and finally, the equation
y (4) + 5y ′′′ − 4y ′′ + y = sin(x)
is a fourth order differential equation.
Note that the order does not depend on whether or not you’vegot ordinary or partial derivatives in the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β
is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t)
which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval.
It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that
the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals
andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart
some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Definitions
Solution
A solution to an ordinary differential equation (ODE) on aninterval α < t < β is any function y(t) which satisfies thedifferential equation in question on the interval. It is important tonote that the solutions are often accompanied by intervals andthese intervals can impart some important information aboutthe solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation
is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written
in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note
about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations
isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products
of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and
neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives
occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any power
other than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
Linear Differential Equations
A linear differential equation is any differential equation thatcan be written in the following form.
an(t)y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = g(t)
The important thing to note about linear differential equations isthat there are no products of the function and its derivatives,and neither the function or its derivatives occur to any powerother than the first power.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t)
can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function.
Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and
its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining
if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear.
If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation
cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called
a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation.
In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation.
Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations
since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know
what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has.
These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Classification
The coefficients an(t), ..., a0(t), g(t) can be any function. Onlythe function y(t) and its derivatives are used in determining if adifferential equation is linear. If a differential equation cannot bewritten in the above form, then it is called a non-lineardifferential equation. In the examples above only
cos(y)d2ydx2 − (1 + y)
dydx
+ y3e−y = 0
is a non-linear equation. Note that we can’t classify Newton’sSecond Law equations since we do not know what form thefunction F has. These could be either linear or non-lineardepending on F .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,
here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t)
is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case,
If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation,
the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Differential Equations in Physics
Schrodinger’s Equation.
− h2
2m∇2Ψ + V (r)Ψ = i h ∂Ψ∂t
Is the starting point for non-relativistic Quantum Mechanics,here the solution Ψ(r , t) is the wave function.
In the one-dimensional case, If we substitute into the aboveequation, the proposed solution
Ψ(x , t) = ψ(x)φ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) =
i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
=
constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant =
E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
we get
− h2
2mψ′′(x)φ(t) + V (x)ψ(x)φ(t) = i hψ(x)φ′(t)
Dividing the whole equation by ψ(x)φ(t) we get
− h2
2mψ′′(x)ψ(x)
+ V (x) = i hφ′(t)φ(t)
= constant = E
So we obtain two ODE’s.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First,
The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second, The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First, The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second, The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First, The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second, The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First, The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second,
The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First, The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second, The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First, The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second, The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
First, The Time Independent Schrodinger Equation
− h2
2md2ψ(x)
dt2 + V (x)ψ(x) = Eψ(x)
and second, The Energy Eigenvalue Equation
i hdφ(t)
dt= Eφ(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations for ClassicalElectromagnetism, here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations for ClassicalElectromagnetism, here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set
of fundamental equations for ClassicalElectromagnetism, here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations
for ClassicalElectromagnetism, here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations for ClassicalElectromagnetism,
here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations for ClassicalElectromagnetism, here the solutions E and B
represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations for ClassicalElectromagnetism, here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Maxwell’s equations:
∇ · E =ρ
ε0, (1a)
∇ · B = 0, (1b)
∇× E = −∂B∂t
, (1c)
∇× B = µ0ε0∂E∂t
+ µ0J, (1d)
This is the set of fundamental equations for ClassicalElectromagnetism, here the solutions E and B represent theelectrical and magnetic fields.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum
ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s.
Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t)
making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the vacuum ρ = J = 0. Maxwell Equations can be written as
∇2E = µ0ε0∂2E∂t2 ∇2B = µ0ε0
∂2B∂t2
Each one of these is 3 PDE’s. Consider the one-dimensionalcase for the Electrical Field
∂2E∂x2 = µ0ε0
∂2E∂t2
Now, suppose that E(x , t) = X(x)T (t) making this substitutionwe get
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
=
µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
=
constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant =
a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
X′′(x)T (t) = µ0ε0X(x)T ′′(t)
dividing by X(x)T (t)
X′′(x)X(x)
= µ0ε0T ′′(t)T (t)
= constant = a
Thus, we have to solve a couple of ODE’s
d2X(x)dx2 = aX(x); µ0ε0T ′′(t) = aT (t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation
for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics,
here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t)
is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time.
In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t .
Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force
bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Newton’s Second Law.
md2rdt2 = F (r , t)
Is the fundamental equation for Classical Mechanics, here thesolution r (t) is the position as a function of time. In this casewe have one independent variable, the time t . Solving anyparticular case of a Force bring us immediately to solving anODE.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Consider the Gravity near earth’s surface
md2rdt2 = −mgk
That vector equation is equivalent to the next three ODE’s
md2xdt2 = 0
md2ydt2 = 0
md2zdt2 = −mg
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.1
Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 for x > 0 .
SolutionWe will need the first and second derivatives :
y ′(x) = −32x−5/2 , y ′′(x) = 15
4 x−7/2
Plug these as well as the function into the differential equation:
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.1Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 for x > 0 .
SolutionWe will need the first and second derivatives :
y ′(x) = −32x−5/2 , y ′′(x) = 15
4 x−7/2
Plug these as well as the function into the differential equation:
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.1Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 for x > 0 .
SolutionWe will need the first and second derivatives :
y ′(x) = −32x−5/2 , y ′′(x) = 15
4 x−7/2
Plug these as well as the function into the differential equation:
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.1Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 for x > 0 .
SolutionWe will need the first and second derivatives :
y ′(x) = −32x−5/2 , y ′′(x) = 15
4 x−7/2
Plug these as well as the function into the differential equation:
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.1Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 for x > 0 .
SolutionWe will need the first and second derivatives :
y ′(x) = −32x−5/2 , y ′′(x) = 15
4 x−7/2
Plug these as well as the function into the differential equation:
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.1Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 for x > 0 .
SolutionWe will need the first and second derivatives :
y ′(x) = −32x−5/2 , y ′′(x) = 15
4 x−7/2
Plug these as well as the function into the differential equation:
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So,
y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2
does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy
the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation
andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence
is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution.
Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include
the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition
thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ?
I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use
this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere
in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the work
showing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function
would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy
the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
4x2(154 x−7/2) + 12x(−3
2x−5/2) + 3x−3/2 = 0
15x−3/2 − 18x−3/2 + 3x−3/2 = 0
0 = 0
So, y(x) = x−3/2 does satisfy the differential equation andhence is a solution. Why then didn’t I include the condition thatx > 0 ? I did not use this condition anywhere in the workshowing that the function would satisfy the differential equation.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form
it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw
in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy
a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation,
because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution
we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues
of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence,
must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
To see why recall that
y(x) = x−3/2 = 1x3/2
In this form it is clear that we will need to avoid x = 0 at theleast as this would give division by zero.
So, we saw in the last example that even though a function maysymbolically satisfy a differential equation, because of certainrestrictions brought about by the solution we cannot use allvalues of the independent variable and hence, must make arestriction on the independent variable.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example,
note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions
to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation.
For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In the last example, note that there are in fact many morepossible solutions to the differential equation. For instance all ofthe following are also solutions
y(x) = x−1/2
y(x) = 5x−3/2
y(x) = 3x−1/2
y(x) = 2x−1/2 + 3x−3/2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up
with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact
an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions
to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So,
given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given
that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are
an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number
of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions
to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation
in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example,
we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask
a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion.
Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is
the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution
that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or
does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matter
which solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution
we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use?
This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question
leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us
to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial
in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
I’ll leave the details to you to check that these are in factsolutions. Given these examples can you come up with anyother solutions to the differential equation?
y(x) = c1x−1/2 + c2x−3/2
There are in fact an infinite number of solutions to thisdifferential equation.
So, given that there are an infinite number of solutions to thedifferential equation in the last example, we can ask a naturalquestion. Which is the solution that we want or does it matterwhich solution we use? This question leads us to the nextmaterial in this section.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2
Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2
is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0
with initial conditions y(4) = 18 and
y ′(4) = − 364 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions
y(4) = 18 and
y ′(4) = − 364 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 and
y ′(4) = − 364 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
Solution
As we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example
the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2
is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and
we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.2Show that y(x) = x−3/2 is a solution to4x2y ′′ + 12xy ′ + 3y = 0 with initial conditions y(4) = 1
8 andy ′(4) = − 3
64 .
SolutionAs we saw in previous example the function y(x) = x−3/2 is asolution and we can then note that
y(4) = 4−3/2 =1
43/2 =18
y ′(4) = −32
4−5/2 = −32
145/2 = − 3
64
and so this solution also meets the initial conditions.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2
is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution
to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation
that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies
these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two
initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value Problem
An Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem
(or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP)
is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation
alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith
an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number
of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3
The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following
is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of
IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
In fact, y(x) = x−3/2 is the only solution to this differentialequation that satisfies these two initial conditions.
Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation alongwith an appropriate number of initial conditions.
Example 1.3The following is an example of IVP
4x2y ′′ + 12xy ′ + 3y = 0, y(4) = 18 , and y ′(4) = − 3
64 .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4
This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is
another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of
an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice,
the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required
willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of Validity
The interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity
for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP
with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.4This is another example of an IVP
2ty ′ + 4y = 3, y(1) = −4 .
As you can notice, the number of initial conditions required willdepend on the order of the differential equation.
Interval of ValidityThe interval of validity for an IVP with initial condition(s)
y(t0) = y0, y ′(t0) = y1, ..., yk (t0) = yk
.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is
the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest
possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval
on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which
the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution
is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid
andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 .
These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define,
but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be
very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficult
to find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find
in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General Solution
The general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution
to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation
is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form
that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution
can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and
doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take
anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions
into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
is the largest possible interval on which the solution is valid andcontains t0 . These are easy to define, but can be very difficultto find in the practice.
General SolutionThe general solution to a differential equation is the mostgeneral form that the solution can take and doesn’t take anyinitial conditions into account.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2
is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution
to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!!
In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact,
all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions
to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equation
will be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form.
This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one
of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equations
that we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn
how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve
and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able
to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify this
shortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.5
y(t) = 34 + c
t2 is the general solution to the equation
2ty ′ + 4y = 3
Check it out !!!!!! In fact, all solutions to this differential equationwill be in this form. This is one of the first differential equationsthat we will learn how to solve and we will be able to verify thisshortly.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular Solution
The particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution
to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation
is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution
that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only
satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation,
but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but also
satisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given
initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6
What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is
the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution
to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Particular SolutionThe particular solution to a differential equation is the specificsolution that not only satisfies the differential equation, but alsosatisfies the given initial condition(s).
Example 1.6What is the particular solution to the following IVP?
2ty ′ + 4y = 3 y(1) = −4
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
This is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually
easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do
than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might,
the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution
isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need
is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine
the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c
that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give us
the solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution
that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after.
To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this,
all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do
isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition
as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionThis is actually easier to do than it might, the general solution isof the form:
y(t) =34+
ct2
All what we need is to determine the value of c that will give usthe solution that we are after. To find this, all we need to do isuse our initial condition as follows:
−4 =34+
c12
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
c = −4− 34= −19
4
So, the particular solution to the IVP is:
y(t) =34− 19
4t2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
c = −4− 34= −19
4
So, the particular solution to the IVP is:
y(t) =34− 19
4t2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
c = −4− 34= −19
4
So, the particular solution to the IVP is:
y(t) =34− 19
4t2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
c = −4− 34= −19
4
So, the particular solution to the IVP is:
y(t) =34− 19
4t2
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit Solution
An explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution
is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution
that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution
is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is,
y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible
to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and
particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7
Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution
of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Implicit/Explicit SolutionAn explicit solution is any solution that is given in the formy = y(t).
An implicit solution is any solution that is not in explicit form,that is, y can not be expressed explicitly as a function of t .Note that it is possible to have either general implicit/explicitsolutions and particular implicit/explicit solutions.
Example 1.7Show that y2 = t2 − 3 is an implicit solution of the differentialequation yy ′ = t .
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
Using implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation,
the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8
Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionUsing implicit derivation, the solution follows:
2yy ′ = 2t + 0
yy ′ = t
Example 1.8Find a particular explicit solution to the IVP
yy ′ = t y(2) = −1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
We already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know
from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example
that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is
y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3.
To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution
allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here.
There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and
in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in fact
only one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one
will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!
We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine
the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function
by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying
the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
SolutionWe already know from the previous example that an implicitsolution to this IVP is y2 = t2 − 3. To find the explicit solution allwe need to do is solve for y(t) from
y(t) = ±√
t2 − 3
Now, we’ve got a problem here. There are two functions hereand we only want one and in factonly one will be the correct !!!We can determine the correct function by reapplying the initialcondition.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case
we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that
the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution,
will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one.
The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find
an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution
to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation.
It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however
that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways
be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible
to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find
an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Only one of them will satisfy the initial condition.
In this case we can see that the negative solution, will be thecorrect one. The explicit solution is then
y(t) = −√
t2 − 3
In this case we were able to find an explicit solution to thedifferential equation. It should be noted however that it will notalways be possible to find an explicit solution.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area.
Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation
for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t
is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and
the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t
is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t),
then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporates
at a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area
can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.9
A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.
Solution
If the volume at time t is denoted by V (t) and the surface attime t is denoted by S(t), then the fact that raindrop evaporatesat a rate proportional to its surface area can be write as
dV (t)dt
proportional to S(t)
dV (t)dt
= −αS(t)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality.
But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒
r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) =
4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 =
− 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 =
− 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c
is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
were α is a constant of proportionality. But, we have thefollowing relationship
V =43
πr3 =⇒ r = [3V4π
]1/3
this implies that
dV (t)dt
= −αS(t) = 4πr2 = − 4απ[3V4π
]2/3 = − 4απ[3
4π]2/3V 2/3
dV (t)dt
= −cV 2/3
where c is a constant.
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug
is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient.
Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug
enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h.
The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues
or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstream
at a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present,
with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug
is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributed
throughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,
write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation
for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present
in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Example 1.10
A certain drug is being administered intravenously to a hospitalpatient. Fluid containing 5mg/cm3 of the drug enters thepatientâAZs bloodstream at a rate of 100cm3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstreamat a rate proportional to the amount present, with a rateconstant of 0.4(h)−1.
(a) Assuming that the drug is always uniformly distributedthroughout the bloodstream,write a differential equation for theamount of the drug that is present in the bloodstream at anytime.
(b) How much of the drug is present in the bloodstream after along time?
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug
is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed
throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream,
and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of
the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg
at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t
isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then
the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation
that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use
is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
=
drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering −
drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exiting
dQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
=
(concentration)(rate of entering)− (concentration)(rate of exiting)
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
Solution
(a) If the drug is always uniformly distributed throughout thebloodstream, and the amount of the drug in mg at time t isdenoted by Q(t), then the equation that we will use is abalance equation
Rate of change of Q(t) =
Rate at which Q(t) enters the bloodstream -
Rate at which Q(t) exits the bloodstream
dQdt
= drug entering − drug exitingdQdt
= (concentration)(rate of entering)− (concentration)(rate of exiting)Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
=
(5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
dQdt
= (5)(100)−Q(t)(0.4)
and the final equation is
dQdt
= 500−Q(t)(0.4)
(b)
Q′(t) = 500−Q(t)(0.4)
Q′(t)500−Q(t)(0.4)
= 1
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ =
ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =
500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4
Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ...
500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
Ordinary differential equationsDefinitionsClassificationBasic Examples
Basic Examples
− ddt
ln|500−Q(t)(0.4)|/0.40 = 1
ln|500−Q(t)(0.4)| = −0.4t + c′
500−Q(t)(0.4) = e−0.4t+c′ = ce−0.4t
Q(t) =500− ce−t
0.4Thus, in the long run the amount of drug present is ... 500/0.4 =20 mg !!!
Dr. Marco A Roque Sol Computational Neuroscience. Session 1-2
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