A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations

TL;DR

This paper introduces a method to derive integrability quantifiers for third-order nonlinear ODEs using Darboux polynomials, simplifying the process of establishing integrability and finding solutions.

Contribution

It presents a novel approach to extract Prelle-Singer quantifiers directly from Darboux polynomials, avoiding solving complex determining equations.

Findings

Successfully derived integrability conditions for various cases

Proved the method's effectiveness through three examples

Facilitated the derivation of general solutions

Abstract

In this paper, we present a method of deriving extended Prelle-Singer method's quantifiers from Darboux Polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and its cofactors, we extract the extended Prelle-Singer method's quantities without evaluating the Prelle-Singer method's determining equations. We consider three different cases of known Darboux polynomials. In the first case, we prove the integrability of the given third-order nonlinear equation by utilizing the Prelle-Singer method's quantifiers from the two known Darboux polynomials. If we know only one Darboux polynomial, then the integrability of the given equation will be dealt as case $2$. Likewise, case $3$ discuss the integrability of the given system where we have two Darboux polynomials and one set of Prelle-Singer method quantity. The established interconnection not only helps in deriving the integrable quantifiers without solving the underlying determining equations. It also provides a way to prove the complete integrability and helps us in deriving the general solution of the given equation. We demonstrate the utility of this procedure with three different examples.