Factorization method for some inhomogeneous Lienard equations

TL;DR

This paper introduces a factorization method to find closed-form solutions for certain inhomogeneous Lienard equations, identifying forcing terms that make these equations integrable and revealing solutions with rational parts and singularities.

Contribution

The paper develops a novel factorization approach that transforms the problem into a system of first-order equations, enabling the explicit solution of inhomogeneous Lienard equations with specific forcing terms.

Findings

Identified conditions for integrability of inhomogeneous Lienard equations.

Derived explicit solutions with rational parts and singularities.

Provided illustrative examples demonstrating the method's effectiveness.

Abstract

We obtain closed-form solutions of several inhomogeneous Lienard equations by the factorization method. The two factorization conditions involved in the method are turned into a system of first-order differential equations containing the forcing term. In this way, one can find the forcing terms that lead to integrable cases. Because of the reduction of order feature of factorization, the solutions are simultaneously solutions of first-order differential equations with polynomial nonlinearities. The illustrative examples of Lienard solutions obtained in this way generically have rational parts, and consequently display singularities.