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1.1 Definitions and Terminology
1.2 Initial-Value Problems 1.3 Differential Equation as Mathematical Models
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DEFINITION: differential equation
An equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE) .
(Zill, Definition 1.1, page 6.)school.edhole.com
Recall Calculus
Definition of a Derivative
If , the derivative of orWith respect to is defined as
The derivative is also denoted by or
)(xfy y )(xfx
h
xfhxf
dx
dyh
)()(lim
0
dx
dfy ,' )(' xf
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Recall the Exponential function
dependent variable: y independent variable: x
xexfy 2)(
yedx
xde
dx
ed
dx
dy xxx
22)2()( 22
2
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Differential Equation: Equations that involve dependent
variables and their derivatives with respect to the
independentvariables.
Differential Equations are classified by
type, order and linearity.
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Differential Equations are classified by
type, order and linearity.
TYPEThere are two main types of differential equation: “ordinary”
and “partial” .
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Ordinary differential equation (ODE)
Differential equations that involve only ONE independent variable are called ordinary differential equations.
Examples:
, , and
only ordinary (or total ) derivatives
xeydx
dy5 06
2
2
ydx
dy
dx
yd yxdt
dy
dt
dx 2
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Partial differential equation (PDE)
Differential equations that involve two or more independent variables are
called partial differential equations.Examples:
and
only partial derivatives
t
u
t
u
x
u
22
2
2
2
x
v
y
u
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ORDERThe order of a differential equation is the order of the highest derivative found in the DE.
second order first order
xeydx
dy
dx
yd
45
3
2
2
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0),(),( dyyxNdxyxM
xeyxy 2'
3'' xy
0),,( ' yyxFfirst order
second order
0),,,( ''' yyyxF
Written in differential form:
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LINEAR or NONLINEAR
An n-th order differential equation is said to be linear if the function
is linear in the variables
there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.
A nonlinear ODE is one that is not linear, i.e. does not have the above form.
)1(' ,..., nyyy
)()()(...)()( 011
1
1 xgyxadx
dyxa
dx
ydxa
dx
ydxa
n
n
nn
n
n
0),......,,( )(' nyyyxF
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LINEAR or NONLINEAR
orlinear first-order ordinary differential equation
linear second-order ordinary differential equation
linear third-order ordinary differential equation
0)(4 xydx
dyx
02 ''' yyy
04)( xdydxxy
xeydx
dyx
dx
yd 53
3
3
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LINEAR or NONLINEAR
coefficient depends on ynonlinear first-order ordinary differential equation
nonlinear function of ynonlinear second-order ordinary differential equation
power not 1 nonlinear fourth-order ordinary differential equation
0)sin(2
2
ydx
yd
xeyyy 2)1( '
024
4
ydx
yd
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LINEAR or NONLINEAR
NOTE:
...!7!5!3
)sin(753
yyy
yy x
...!6!4!2
1)cos(642
yyy
y x
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Solutions of ODEsDEFINITION: solution of an ODEAny function , defined on an interval I and possessing at least n derivatives that are continuouson I, which when substituted into an n-th order ODE reduces the equation to an identity, is said to be a solution of the
equation on the interval .)Zill, Definition 1.1, page 8.(
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Namely, a solution of an n-th order ODE is a function which possesses at least nderivatives and for which
for all x in I
We say that satisfies the differential equation on I.
0))(),(),(,( )(' xxxxF n
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Verification of a solution by substitution Example:
left hand side:
right-hand side: 0
The DE possesses the constant y=0 trivial solution
xxxx exeyexey 2, '''
xxeyyyy ;02 '''
0)(2)2(2 ''' xxxxx xeexeexeyyy
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DEFINITION: solution curveA graph of the solution of an ODE is called a solution curve, or an integral curve
of the equation.
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DEFINITION: families of solutions A solution containing an arbitrary constant
(parameter) represents a set ofsolutions to an ODE called a one-parameter
family of solutions. A solution to an n−th order ODE is a n-
parameter family of solutions .
Since the parameter can be assigned an infinite number of values, an ODE can have an infinite number of solutions.
0),,( cyxG
0),......,,( )(' nyyyxF
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Verification of a solution by substitution Example: 2' yy
xkex 2)(φ2' yy
xkex 2)(φxkex )(φ '
22)(φ)(φ ' xx kekexx
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Figure 1.1 Integral curves of y’+ y = 2 for k = 0, 3, –3, 6, and –6.
©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
xkey 2
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Verification of a solution by substitution Example:
1' x
yy
Cxxxx )ln()(φ 0x
Cxx 1)ln()(φ '
1)(φ
1)ln(
)(φ '
x
x
x
Cxxxx
for all ,
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 1.2 Integral curves of y’ + ¹ y = ex
for c =0,5,20, -6, and –10. x
)(1
cexex
y xx
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Example: is a solution of
By substitution:
)4sin(17)4cos(6)(φ xxx 016'' xy
0φ16φ
)4sin(272)4cos(96φ
)4cos(68)4sin(24φ
''
''
'
xx
xx
0),,,( ''' yyyxF
0)(φ),(φ),(φ, ''' xxxxFschool.edhole.com
Consider the simple, linear second-order equation
,
To determine C and K, we need two initial conditions, one specify a point lying on the solution curve and the other its slope at that point, e.g. ,
WHY ???
012'' xy
xy 12'' Cxxdxdxxyy 2'' 612)('
KCxxdxCxdxxyy 32' 2)6()(
Cy )0('Ky )0(
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IF only try x=x1, and x=x2
It cannot determine C and K,
KCxxxy
KCxxxy
23
22
13
11
2)(
2)(
xy 12'' KCxxy 32
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 2.1 Graphs of y = 2x³ + C x +K for various values of C and K.
e.g. X=0, y=k
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 2.2 Graphs of y = 2x³ + C x + 3 for various values of C.
To satisfy the I.C. y(0)=3The solution curve must pass through (0,3)
Many solution curves through (0,3)
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 2.3 Graph of y = 2x³ - x + 3.
To satisfy the I.C. y(0)=3,y’(0)=-1, the solution curve must pass through (0,3)having slope -1
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Solutions General Solution: Solutions obtained from integrating the differential equations are called general solutions. The general solution of a nth order ordinary differential equation contains n arbitrary constants resulting from integrating
times. Particular Solution: Particular solutions are the solutions obtained by assigning specific values to
the arbitrary constants in the general solutions. Singular Solutions: Solutions that can not be expressed by the general solutions are called
singular solutions. school.edhole.com
DEFINITION: implicit solutionA relation is said to be an implicit solution of an ODE on an interval I provided there exists at least one function that satisfies the relation as well as the differential equation
on I .a relation or expression that defines a solution implicitly.
In contrast to an explicit solution
0),( yxG
)(xy
0),( yxG
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DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation
Cyxxxyy 232 22
0)22(34 ' yyxxy
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DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation Cyxxxyy 232 22
0)22(34 ' yyxxy
0)22(34
02234
02342
/)(/)232(
'
'''
'''
22
yyxxy
yyyxyxy
yxxyyyy
dxCddxyxxxyyd
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Conditions Initial Condition: Constrains that are specified at the initial point, generally time point, are called initial conditions. Problems with specified initial conditions are called initial value
problems.
Boundary Condition: Constrains that are specified at the boundary points, generally space points, are called boundary conditions. Problems with specified boundary conditions
are called boundary value problems. school.edhole.com
First- and Second-Order IVPS Solve:
Subject to:
Solve:
Subject to:
00 )( yxy
),( yxfdx
dy
),,( '2
2
yyxfdx
yd
10'
00 )(,)( yxyyxy school.edhole.com
DEFINITION: initial value problem
An initial value problem or IVP is a problem which consists of an n-th order ordinary differential equation along with n initial conditions defined at a point found in the interval of definition differential equation
initial conditions
where are known constants.
0x
),...,,,( )1(' nn
n
yyyxfdx
yd
10)1(
10'
00 )(,...,)(,)( nn yxyyxyyxy
I
110 ,...,, nyyyschool.edhole.com
THEOREM: Existence of a Unique Solution
Let R be a rectangular region in the xy-plane defined by that contains
the point in its interior. If and are continuous on R, Then there
exists some interval contained in anda unique function defined on that is a solution of the initial value
problem.
dycbxa ,
),( 00 yx ),( yxf
yf /
0,: 000 hhxxhxI
bxa )(xy
0I
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