Exact solutions of coupled Li\'enard-type nonlinear systems using factorization technique

TL;DR

This paper develops a factorization method to find exact solutions of coupled Lie9nard-type nonlinear differential equations, including a class of equations solvable via Bernoulli equations, advancing solution techniques for complex nonlinear systems.

Contribution

It introduces a generic algorithm for solving coupled Lie9nard systems using factorization, including solutions for equations reducible to Bernoulli form.

Findings

Identified a class of coupled equations solvable by Bernoulli equations

Derived general solutions for certain coupled Lie9nard systems

Demonstrated the method on generalized Emden equations

Abstract

General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases particular solutions may be found with little effort if it is possible to factorize the equation in terms of first order differential operators. In our present study we use this factorization technique to address the problem of finding solutions of a system of general two-coupled Li\'enard type nonlinear differential equations. We describe a generic algorithm to identify specific classes of Li\'enard type systems for which solutions may be found. We demonstrate this method by identifying a class of two-coupled equations for which the particular solution can be found by solving a Bernoulli equation. This class of equations include coupled generalization of the modified Emden equation. We further deduce the general solution of a class of coupled ordinary differential equations using the factorization procedure discussed in this manuscript.