Invariant characterization of scalar third-order ODEs that admit the maximal point symmetry Lie algebra

TL;DR

This paper uses Cartan's equivalence method to characterize third-order scalar ODEs with maximal symmetry, providing a systematic way to find transformations reducing them to the simplest linear form.

Contribution

It offers an invariant characterization and auxiliary functions for third-order ODEs with maximal symmetry, facilitating their reduction to linear equations.

Findings

Invariant criteria for maximal symmetry ODEs

Explicit auxiliary functions for reduction

Examples illustrating the method

Abstract

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'"=f(x,u,u',u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation $\bar{u}'"=0$. Moreover, examples are given to illustrate the method.