Sec 6.3 Separation of Variables (Homogeneous Equations)

Read Intro p.423

A function ( , ) is a homogeneous function of degree n if

( , ) ( , )

where n is any real number.

n

f x y

f tx ty t f x y

To test if a function is homogeneous, replace each variable in the equation by t times that variable. Replace x with tx, and y with ty.

Simplify completely.

Attempt to factor out all remaining t’s as a Greatest Common Factor. (GCF)

If this is successful, then the function is homogeneous!

Read Example 4 p.423 for additional example

2 2

Ex1) Test each function to see if it is homogeneous:

a. x 0xy dx y dy

2 2( )( ) 0tx tx ty dx ty dy

2 2 2 2 2 0t x t xy dx t y dy

2 2 2 0t x xy dx y dy

This is f(x,y)

This is f(x,y)2t

2 2. x 1 0b dx y dy

This function is homogeneous

of degree 2.

2 21 0tx dx ty dy

2 2 2 21 0t x dx t y dy This function is NOT homogeneous since I cannot factor t’s out due to the 1!

If a differential equation is homogeneous then it can be transformed into a differential equation whose variables

are separable by the substitutiony=vx

where v is a differentiable function of x.

Note: if y = vx, then dy = v dx + x dv

2 2

Ex5a) continued ... separate the differentiable equation

x 0xy dx y dy

22 x ( ) 0x vx dx xv vdx xdv 2 2 2 3 3 2 x 0dx x vdx x v dx x v dv

3 2 0dx vdx v dx xv dv 3 2 dx vdx v dx xv dv

3 2 1 v v dx xv dv

2

3

1

dx v dv

x v v