On the complete integrability and linearization of nonlinear ordinary differential equations - Part IV: Coupled second order equations

TL;DR

This paper develops a systematic method using the extended Prelle-Singer procedure to solve and linearize coupled second order nonlinear differential equations, including a new transformation technique for uncoupling them.

Contribution

It introduces a comprehensive approach for solving coupled second order nonlinear ODEs and a novel method to transform them into uncoupled equations.

Findings

Derived general solutions for coupled second order nonlinear ODEs.

Presented a new transformation method to uncouple nonlinear ODEs.

Illustrated the methods with important example equations.

Abstract

Coupled second order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations we focus our attention on the method of deriving general solution of two coupled second order nonlinear ordinary differential equations through the extended Prelle-Singer procedure. We describe a procedure to obtain integrating factors and required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above, we introduce a new method of transforming coupled second order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples.