$\lambda$-symmetry criteria for linearization of second order ODEs via point transformations

TL;DR

This paper introduces a new $-symmetry linearization criterion for second order ODEs, simplifying the process of finding linearization transformations by leveraging the relationship between $-symmetries and first integrals.

Contribution

It presents a novel $$-symmetry based linearization criterion that reduces complexity in constructing linearization transformations for second order ODEs.

Findings

Derived a new $$-symmetry linearization criterion.

Successfully obtained local linearization transformations for specific nonlinear ODEs.

Illustrated the approach with examples of quadratic and cubic first derivative nonlinear ODEs.

Abstract

An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $\lambda$-symmetries and first integrals. This relation further leads to a new $\lambda$-symmetry linearization criteria for second order ODEs which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ODEs of the form $y''+F_1(x,y)y'+F(x,y)=0$. Examples of linearizing nonlinear ODEs which are quadratic or cubic in the first derivative are also presented.