The Lie symmetry group of the general Lienard-type equation

Abstract

We consider the general Lienard-type equation $\ddot{u}={\sum }_{k=0}^n{f}_k{\dot{u}}^k$ for $n\ge 4$. This equation naturally admits the Lie symmetry $\frac{\partial }{\partial t}$. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions $f_0,\dots ,f_n$. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases $n=1$ or $n=2$. That is, this paper handles all remaining cases except for $n=3$.