A class of exact solutions of the Li\'enard type ordinary non-linear differential equation

TL;DR

This paper derives a class of exact parametric solutions for Lie9nard type nonlinear differential equations by transforming them into Abel equations and applying an extended Chiellini integrability condition.

Contribution

It introduces a method to obtain exact solutions for Lie9nard equations via transformation to Abel equations and extends the Chiellini integrability condition to general Abel equations.

Findings

Exact parametric solutions for Lie9nard equations are derived.

The Chiellini integrability condition is extended to general Abel equations.

Explicit solutions for generalized van der Pol equations are provided.

Abstract

A class of exact solutions is obtained for the Li\'{e}nard type ordinary non-linear differential equation. As a first step in our study the second order Li\'{e}nard type equation is transformed into a second kind Abel type first order differential equation. With the use of an exact integrability condition for the Abel equation (the Chiellini lemma), the exact general solution of the Abel equation can be obtained, thus leading to a class of exact solutions of the Li\'{e}nard equation, expressed in a parametric form. We also extend the Chiellini integrability condition to the case of the general Abel equation. As an application of the integrability condition the exact solutions of some particular Li\'{e}nard type equations, including a generalized van der Pol type equation, are explicitly obtained.