Differential Equations Any equation containing a derivative is called a differential equation.

Differential Equations

Any equation containing a derivative is called a differential equation.

Differential Equations

A function which satisfies the equation is called a solution to the differential equation.

Differential Equations

The general solution is a family of curves. Each member is a particular solution

.

Differential Equations

The order of the differential equation is the order of the highest derivative involved.

These are examples of FIRST ORDER differential equations

Differential Equations

The order of the differential equation is the order of the highest derivative involved.

These are examples of SECOND ORDER differential equations

3 Types of Differential Equations

Example:

3 Types of Differential Equations

Example:

3 Types of Differential Equations

Example:

3 Types of Differential Equations

Example:

This is the general solution

The constant can be written as part of the logarithm

Example:

Particular solution

Solve:

If when

Particular solution

Solve:

If when

Particular solution

Solve:

If when

Particular solution

Solve:

2nd Types of Differential Equations

)(yfdx

dy form the of is equation aldifferenti a When

dxdyyf

)(

1We can then integrate both sides.

dxyf

dy

)(This will obtain the general solution.

Example

A curve passes through the point (0, π/3) and its gradient at the point (x, y) is given by

Find the equation of the curve. (Sidebotham)

General Solution

A curve passes through the point (0, π/3) and its gradient at the point (x, y) is given by

Find the equation of the curve. (Sidebotham)

General Solution

A curve passes through the point (0, π/3) and its gradient at the point (x, y) is given by

Find the equation of the curve. (Sidebotham)

When the equation is of the form ( ) ( ), thendy

f x g ydx

1( )

( )dy f x dx

g y

( )( )dy

f x dxg y

Type 3

Solve the differential equation 2 3dy

xy xdx

Solve the differential equation 2 3dy

xy xdx

Solve the differential equation 2 3dy

xy xdx

Testing solutions

• Prove that is a solution to the differential equation

•

Do not try to integrate

•

Do not try to integrate

• Substitute