What are homogeneous differential equations?

Overview

The differential equations having function of the same degree are said to be homogeneous differential equations.

Differential equation is an equation involving an unknown function and its derivatives.

Ex.

“A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable.”

Notation

The expression y’, y’’ , y’’’ …….yⁿ are used to represent derivatives.

Here,  y’ =       ,     y’’ =       and so on.

Homogeneous differential equations

A first order differential equation is homogeneous. It can be written in the form as,

And each term of  and  has same degree.

Solving a homogeneous differential equation

If the differential equation is homogeneous, we put

y = vx, and dy/dx = v + x dv/dx

We follow the Steps given below to solve a homogeneous differential equation-

1. Put y = vx and dy/dx = v + x dv/dx

2. Separate the variable

3. Integrate on both sides

4. Now put y = v/x and solve.

Solved examples

Example:  solve the homogeneous differential equation given below:

Solution:

 Here the function is –

 We will put  y = vx as mentioned above –

dy/dx = v + x dv/dx

 Separate the variables –

dv / sin v = dx

cosec v dv = dx

On integrating –

Example: Solve the following homogeneous differential equation-

Solution:

On solving, we get-

Now separating the variable, we get-

On integrating both sides, we obtain –

Or

Interested in learning about similar topics? Here are a few hand-picked blogs for you!

• What is Cayeley-Hamilton theorem?