Homogeneous Order

Homogeneous Order

Within a polynomial of two variables, the degree of each term is the sum of the degrees (powers) of the individual variables. For example

If all of the terms of a polynomial have the same degree, we say the polynomial is homogeneous. For example, the following polynomials are homogeneous

If a first order differential equation is of the form

where all of the terms of and are of the same degree, then we can use a substitution to transform the equation into a separable first order differential equation.

The substitution we use is

(**)

which completes the terms we need to carry out the substitution.

Example: Find the general solution of

First we divide the top and bottom of the right hand side fraction by

then make the substitution (*) and (**) to get

Then isolate the derivative term:

The equation is now separable so we can separate and integrate

STEP 2013, Math I, #7: (i) Use the substitution

, where is a function of , to show that the solution of the

differential equation

that satisfies when is

( )

(ii) Use a substitution to find the solution of the differential equation

that satisfies when .

STEP 2012, Math I, #8: (i) Show that substituting

, where is a function of , in the differential equation

leads to the differential equation

.

Hence show that the general solution can be written in the form

where

is a constant.

(ii) Find the general solution of the differential equation