New classification techniques for ordinary differential equations

TL;DR

This paper introduces an enhanced ODE solver leveraging Cartan's equivalence method, which finds transformations to well-known equations without integration, improving efficiency and theoretical understanding.

Contribution

It proposes a novel ODE solving approach using Cartan's equivalence method, linking coordinate changes to symmetry pseudo-groups, and utilizes Kamke's ODE dictionary for classification.

Findings

Transforms to known ODEs without integration

Establishes relationship between coordinate changes and symmetry pseudo-groups

Provides a theoretical framework for equivalence transformations

Abstract

The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of \'Elie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODE, that are regarded as well-known, or at least well-studied. The dictionary considered in this article are ODE in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.