Self-equivalence 3rd order ODEs by time-fixed transformations

TL;DR

This paper investigates the classification of third-order ordinary differential equations under time-fixed coordinate transformations using Cartan's equivalence method, aiming to identify when two such equations are equivalent.

Contribution

It applies Cartan's equivalence method to third-order ODEs with fixed time transformations, providing a systematic approach to their local equivalence classification.

Findings

Derived criteria for local equivalence of 3rd order ODEs

Established invariants under time-fixed transformations

Enhanced understanding of the geometric structure of these equations

Abstract

Let y''' = f(x, y, y', y'') be a 3rd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates.