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

In this paper, we present a method for deriving quantifiers of the extended Prelle–Singer (PS) method using Darboux polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and their co-factors, we extract the extended PS method’s quantities without evaluating the PS 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 utilising the quantifiers of the PS method 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 discusses the integrability of the given system where we have two Darboux polynomials and one set of PS method quantity. The established interconnection not only helps in deriving the integrable quantifiers without solving the underlying determining equations, but 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.