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

TL;DR

This paper classifies Lie9nard equations with maximal Lie point symmetries, showing they are linearizable and deriving their transformations and solutions, thus extending understanding of their symmetry properties.

Contribution

It identifies conditions under which Lie9nard equations have maximal Lie symmetries and provides explicit linearizing transformations and solutions.

Findings

Equations with maximal Lie symmetries are linearizable.

Maximal symmetry occurs only when second derivative of f(x) is zero.

Explicit linearizing transformations and solutions are derived.

Abstract

In this second of the set of two papers on Lie symmetry analysis of a class of Li\'enard 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 Li\'enard 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.