On the local equivalence of partial differential equations

TL;DR

This paper explores the concept of local equivalence in partial differential equations, emphasizing its importance in solving and understanding PDEs, inspired by Cartan's 1914 work, and illustrating with examples.

Contribution

It introduces a framework for analyzing local equivalence of PDEs, connecting integration methods to equivalence problems, and revisiting Cartan's classical insights.

Findings

Local equivalence provides a unifying perspective for PDE solutions.

Examples demonstrate how equivalence simplifies complex PDE analysis.

The approach links classical methods to modern geometric theory.

Abstract

In all the practical applications of partial differential equations, what is mostly needed and what is in fact hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on a most enlightening M\'emoire written by \'Elie Cartan in 1914 and that the majority ignores. We discuss a setting for the local equivalence problem and illustrate it by some examples. It should also be noted that any integration process or method is in fact a local equivalence problem involving a suitable model.