Oxford Calculus: Solving Homogeneous First Order Differential Equations

University of Oxford mathematician Dr Tom Crawford explains how to solve homogeneous first order differential equations with a worked through example involving partial fractions and implicit differentiation.

Test yourself with some exercises on solving homogeneous first order differential equations with this FREE worksheet in Maple Learn here.

We start by introducing the general form of a homogeneous first order differential equation and the substitution required to solve problems of this type. The resulting equation can then be solved using the method of ‘separation of variables’.

The method of solution is then demonstrated with a fully worked example. Once the change of variables has been made, the resulting integral is solved using partial fractions and the reverse chain rule. The answer is then checked by implicitly differentiating and rearranging to get the required form seen in the original equation.


  1. Tom — Thank you very much for the excellent videos.

    Can the ode you selected be derived from an actual physical phenomena?

    Maybe population dynamics?


    • This example I’m not sure, but there are lots of population models and chemical reaction kinetics models that give rise to the ‘rational polynomial’ type ODEs that we see here for sure.


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