Linearization of third-order ordinary differential equations u'''=f(x,u,u',u'') via point transformations

TL;DR

This paper fully characterizes third-order ODEs that can be linearized via point transformations using Cartan's method, providing invariants, a construction procedure, and practical applications.

Contribution

It completes the classification of linearizable third-order ODEs by deriving invariant conditions and a systematic method for finding the linearizing transformations.

Findings

Invariant characterization of linearizable third-order ODEs.

A procedure to construct canonical forms from invariants.

Applications demonstrating the method's effectiveness.

Abstract

The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a four-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 invariant is also presented. The method provides auxiliary functions which can be utilized to efficiently determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.