On the local equivalence of partial differential equations

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.