A Group Theoretical Identification of Integrable Equations in the Liénard Type Equation $\ddot{x}+f(x)\dot{x}+g(x) = 0$ : Part II: Equations having Maximal Lie Point Symmetries

Abstract

In this second of the set of two papers on Lie symmetry analysis of a class of Liénard type equation of the form $\ddot {x} + f(x)\dot {x} + g(x)= 0$, where over dot denotes differentiation with respect to time and $f(x)$ and $g(x)$ are smooth functions of their variables, we isolate the equations which possess maximal Lie point symmetries. It is well known that any second order nonlinear ordinary differential equation which admits eight parameter Lie point symmetries is linearizable to free particle equation through point transformation. As a consequence all the identified equations turn out to be linearizable. We also show that one can get maximal Lie point symmetries for the above Liénard equation only when $f_{xx} =0$ (subscript denotes differentiation). In addition, we discuss the linearising transformations and solutions for all the nonlinear equations identified in this paper.