Exact solutions of coupled Lienard-type nonlinear systems using factorization technique

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 Lienard-type nonlinear differential equations. We describe a generic algorithm to identify specific classes of Lienard-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 ODEs using the factorization procedure discussed in this paper.