# Homogeneous

The solution of homogeneous differential equations including the use of the D operator

**Contents**

## Definition

The equation is said to be**homogeneous**if P and Q are homogeneous functions of and of the same degree.

For example :

If We can test to see whether this first order equation is homogeneous by substituting . If the result is in the form i.e. all the 's are canceled then the test is satisfied and the equation is Homogeneous.

Example:

##### Example - Simple example

Problem

Workings

Becomes

Solution

There are no terms in on the right hand side and the equation is Homogereous.

So the original equation is

**not**homogeneous.## The General Form Of A Homogeneous Linear Equation

A

For example : is homogeneous polynomial .

**homogeneous**polynomial is a polynomial whose monomials with nonzero coefficients all have the same total degree.For example : is homogeneous polynomial .

## The Use Of The D Operator To Solve Homogeneous Equations

If andThe

**D operator**is a linear operator defined as : . For example :Example:

##### Example - A complex example

Problem

Solve the following Differential equation:

Workings

By putting and using the D factor then the equation reduces to:-
Or
Therefore
Therefore

Solution

Therefore

## Equations Which Can Be Reduced To The Homogeneous Form

Consider the following equation: The equation is not Homogeneous due to the constant terms and However if we shift the origin to the point of intersection of the straight lines and , then the constant terms in the differential equation will disappear.Example:

##### Example - Rational Example

Problem

Workings

The lines and meet at the point (1, 2). We therefore make the following substitutions:
The equation now becomes:

Solution

This is homogeneous and can be solved by putting Y = v X. The solution is given by:

## Exceptional Case

If the two straight lines are parallel, then there is no finite point of intersection and we proceed as follows:Let
Put
Then
Thus the equation becomes:
Therefore
Therefore
Therefore
Thus

Last Modified: 31 Dec 11 @ 00:41 Page Rendered: 2022-03-14 15:51:55