# Differential Equations, Homogeneous Differential Equations, Solution of Homogeneous Differential Equation

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## Homogeneous Differential Equations:

Consider the following differential equations:

In equation (i) above, we see that each term except is of degree

[As degree of is , degree of is and degree of is ]

In equation (ii) each term except is of degree.

In equation (iii) each term except is of degree, as it can be rewritten as

Such equations are called homogeneous equations.

**Note:**

Homogeneous equations do not have constant terms.

## Solution of Homogeneous Differential Equation:

To solve such equations, we proceed in the following manner:

I. Write one variable. (the other variable).

(i.e. either or )

II. Reduce the equation to separable form

III. Solve the equation as we had done earlier.

Example:

Solve:

Solution:

The given differential equation is

Or

It is a homogeneous equation of degree two.

Let . Then

From (1), we have

or

Or or

Or or

Or

Further on integrating both sides of (2), we get

where is an arbitrary constant.

On substituting the value of , we get

which is the required solution.

**Note:**

If the Homogeneous differential equation is written in the form then is substituted to find solution