Equivalence Problem for Non-Linearizable Third-Order ODEs with Four-Dimensional Lie Symmetry Subalgebras under Point Transformations

TL;DR

This paper uses Cartan's method to classify non-linearizable third-order ODEs with four-dimensional Lie symmetry subalgebras and provides a way to find point transformations that relate them.

Contribution

It explicitly constructs invariant coframes for these ODEs and introduces a method to derive point transformations from them.

Findings

Explicit invariant coframes for four branches of ODEs

A systematic method to construct point transformations

Illustrative examples demonstrating the approach

Abstract

Cartan's equivalence method is applied to explicitly construct invariant coframes for four branches, which are used to characterize all non-linearizable third-order ODEs with a four-dimensional Lie symmetry subalgebra under point transformations. Additionally, we present a method for constructing the point transformations based on the derived invariant coframes. Examples are provided to illustrate our approach.