Invariant characterization of third-order ODEs $u'''=f(x,u,u',u'')$ that admit a five-dimensional point symmetry Lie algebra

TL;DR

This paper uses Cartan's method to characterize third-order ODEs with a five-dimensional symmetry algebra, providing invariant criteria, a canonical form construction, and methods for reduction to linear form.

Contribution

It offers a new invariant characterization of third-order ODEs with five-dimensional symmetry algebras using Cartan's equivalence method, including procedures for canonical form and linearization.

Findings

Invariant criteria for five-dimensional symmetry algebra

Procedure for constructing canonical forms

Method for reducing equations to linear form

Abstract

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function $f$ in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.