In chapters 5 and 11 we have seen that any second order PDE S ⊂ J2 admits two Monge characteristic vector field systems. S is said to be hyperbolic if these are different, and parabolic if they coincide. Vessiot's study of those hyperbolic PDEs for which each Monge system admits at least two functionally independent first integrals led to a remarkable classification, presented in chapters 12 and 13.

The corresponding classification of those parabolic PDEs for which the double Monge system admits at least two functionally independent first integrals is the subject of Cartan's ‘five variable paper’ [Cartan 1910].

While Vessiot's classification is based upon a straightforward stepwise simplification of the Monge systems, Cartan instead starts by solving a very general equivalence problem, which he then applies to the special case of parabolic PDEs.

Cartan's ideas for solving the equivalence problem were first announced in the Comptes Rendus note [Cartan 1902]. In order to make this comprehensible he then went on with his general theory of Lie pseudogroups in [Cartan 1904] and [Cartan 1905], whereupon the first chapter of [Cartan 1908] is devoted to a full explanation of the equivalence problem. And this is then applied to the classification of parabolic PDEs in [Cartan 1910].

This chapter is a survey of Cartan's solution of the equivalence problem, mainly following [Cartan 1937a]. For more modern references see e.g. [Gardner 1989], [Morimoto 1993], and [Olver 1995].

A different and very interesting approach to the equivalence problem has been developed by Pommaret, taking [Vessiot 1903] as point of departure—see e.g. [Pommaret 1978] and [Pommaret 1983].