Homogeneous Equations

A function of two variables f(x,y) is said to be homogeneous of degree n if there is a constant n such that

, ,nf tx ty t f x y for all t, x and y.

A differential equation of the form

( , ) ( , ) 0M x y dx N x y dy

Is homogeneous if M and N are both homogeneous functions of the same degree.

To solve homogeneous equations, turn them into separable equations using the substitution: y xv .

Example:

Solve the equation: 2 2 2 0x y dx xydy

Exact Equations

Given a function ,f x y , its total differential is: f f

df dx dyx y

.

Hence, the family of curves ,f x y c satisfy the differential equation

0f f

dx dyx y

So, if there is a function such that ( , )f

M x yx

and ( , )f

N x yy

, then

( , ) ( , ) 0M x y dx N x y dy is called and “exact differential equation”.

How can you decide that a differential equation is exact?

Check: ??M N

y x

Example:

Solve: 3 21 2 4 0xy dx y x dx

Non-Exact Equations and Integrating Factors

Example: 21 0xy x dx x dy

Let ( , ) ( , ) 0M x y dx N x y dy be a non-exact equation.

Case 1: If y xM N

N

is a function of “x” only, call this function ( )f x . The

integrating factor is f x dxx e .

Case 2: If y xM N

M

is a function of “y” only, call this function ( )f y . The

integrating factor is f y dyy e .

Example: Find the integral curve of

2

30

dy x y

dx x y

that passes through the point (1,1).

Linear Equations with Constant Coefficients:

Example:

Find the general solution:

a) '' 4 0y y

b) 2 '' 7 ' 4y y y

c) '' 2 ' 0y y y

d) ''' '' 9 ' 9 0y y y y

Example: Solve: ''y x y

Example: Find the Laplace Transform of 2xf x e .

Formula: 0

( ) ( )stL f e f t dt

.

Euler’s Equations (Can find this on Chapter 3 notes – 3.2)

Rewrite: 2

2 ( 1) 0 where lnd y dy

y z xdzdz

Example:

Solve 2 '' 4 ' 6 0x y xy y