Differential equations

Differential equations

• Many applications of mathematics involve two variables and a relation between them is required.

• This relation is often expressed in terms of the rate of change of one variable with respect to another.

• This leads to a differential equation.• Its solution is an equation connecting the

two variables.

Types of differential equation

We will be solving differential equations of the form:

( ) dy

f xdx

Type 1:

( ) dy

f ydx

Type 2 :

( )

( )

dy f x

dx g yType 3 :

Type 1: example0

1

At each point of a curve for which the tangent cuts the -axis at ,

and is the foot of the perpendicular from the -axis to . If is always unit

below , find the equation of the curve.

P x y T

N y P T

N

1

1

Since , the gradient of the tangent

is , so that

.

NP x

xdy

dx x

0ln ln( ) (since ).After integrating, we have xy x k x k

ln( )

10

The equation is called the general solution of the

differential equation , for .

y x k

dyx

dx x

This solution can be represented by a , or

, one for each value of .k

family of curves

solution curves

Type 1: word problems

Do Q1-Q8, pp.277-278

Type 2 differential equations

1dx dy

dxdy

1( )

( )

1

( )

dy dxf y

dx dy f y

x dyf y

Do Q1-Q7, pp.284-285

Type 3 differential equation

( )( ) ( )

( )

Separable equations

dy f x

g y dy f x dxdx g y

Do Q1-Q12, pp.289-290 Do Q1-Q23, pp.290-295