An alternative proof of Lie’s linearization theorem using a new λ-symmetry criterion

Abstract

An alternative proof of Lie’s approach for the linearization of scalar second-order ordinary differential equations is derived by using the relationship between λ-symmetries and first integrals. This relation further leads to a new λ-symmetry linearization criterion for second-order ordinary differential equations 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 ordinary differential equations of the form y″+F1(x,y)y′+F(x,y)=0. Examples of linearizable nonlinear ordinary differential equations which are quadratic or cubic in the first derivative are also presented.