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# Homogeneous

The solution of homogeneous differential equations including the use of the D operator View other versions (2)

## 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.

## Methods Of Solution

A solution can be found by putting on both sides of the equation:
Example:
##### Example - Rational Example
Problem
Workings
Putting

Since y is a function of x so is v

Therefore
Therefore

Separating the variables

Integrating

But so
Solution
Substituting equation (2) in equation (1)

## The General Form Of A Homogeneous Linear Equation

A homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have the same total degree.
For example : is homogeneous polynomial . The method to solve this is to put and the equation then reduces to a linear type with constant coefficients.
If Then Therefore Also Therefore Hence And ## The Use Of The D Operator To Solve Homogeneous Equations

If and The D operator is a linear operator defined as : . For example : Then from equation (2) And from equation (3) 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 