# Some Homogeneous Ordinary Differential Equations

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows a procedure for solving an ordinary differential equation of the form . The first step is to introduce a new variable , . Differentiating the last equation, we get . By substitution, we get , . In the last equation, we separate variables to get . Integration of both parts yields . From the last equation, we get a general solution of the form where .

Contributed by: Izidor Hafner (April 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The equation is called homogeneous if and are homogeneous functions of of the same order. The equation can be reduced to the form . A function is called homogeneous of order if . An example: and are homogeneous of order 2, and is homogeneous of order 0.

The differential equation is not homogeneous in the usual sense of a linear differential equation having a right-hand side equal to zero, like .

Reference

[1] V. I. Smirnoff, *Lectures in Higher Mathematics* (in Russian), Vol. 2, Moscow: Nauka, 1967 pp. 19–21.

## Permanent Citation