Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations

Abstract

Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations $\ddot{x}+f(x)\dot{x}+x=0$, being $f(x)$ a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.

Key words
Liénard equation; rational first integral; Poincaré problem; polinomial differential equation