Some Homogeneous Ordinary Differential Equations

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 .

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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.
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