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

Differential Equations

Any equation containing a derivative is called a differential equation.


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


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

.


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

These are examples of FIRST ORDER 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:


Example:


Example:


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



General Solution



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


xy xdx


xy xdx

Testing solutions

• Prove that is a solution to the differential equation

•

Do not try to integrate

•

Do not try to integrate

• Substitute

Documents

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