Invariant algebraic curves for certain generalized Liénard differential system

Abstract

In this work we solve the problem of finding the invariant algebraic curves of a generalized Liénard differential system $ \dot{x} = y $, $ \dot{y} = -f(x)y-g(x) $, where $ \deg f = m $ and $ \deg g = n $ and with $ n = m+1 $, generalizing the known previous examples. In particular it is studied the case $ m = 3 $ and $ n = 4 $. The difficulties in applying the Puiseux method are shown even when the degrees of the invariant curves are bounded.