Liouvillian Integrability Versus Darboux Polynomials

Abstract

In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Lienard polynomial differential system \(\dot{x}=y,\, \dot{y}=-f(x)y-g(x)\) with \(\deg f>\deg g\) is not Liouvillian integrable.