© 2016, Springer International Publishing. 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 Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x) with deg f> deg g is not Liouvillian integrable.
- Darboux Jacobian multiplier
- Darboux polynomial
- Liouvillian integrability
- Polynomial differential system