Linearization Problem for Third-Order ODEs with Four- and Five-Dimensional Lie Symmetry Algebras under Contact Transformations

TL;DR

This paper uses Cartan's method to identify and construct contact transformations that linearize third-order ODEs with specific Lie symmetry algebra dimensions, providing a systematic approach and examples.

Contribution

It introduces a Cartan equivalence-based procedure for linearizing third-order ODEs with four- and five-dimensional Lie symmetry algebras under contact transformations.

Findings

Constructed invariant coframes for rank one and zero cases.

Derived explicit contact transformations for linearization.

Provided illustrative examples demonstrating the method.

Abstract

Using Cartan equivalence method, invariant coframes are constructed for two branches of rank one and zero, which characterize linearizable third-order ODEs under contact transformations with four- and five-dimensional Lie symmetry algebras, respectively. A procedure for deriving the corresponding contact transformations is also presented, along with illustrative examples.