ABSTRACT
The two main topics of this chapter, equivalence and invariance of differential equations w.r.t. certain transformation groups, have their origin in the theory of algebraic forms as developed in the 19th century. Comprehensive introductions to the latter may be found in one of the classical text books like Clebsch [30] or Gordan [55], or in the expository article by Kung and Rota [97]. There was a general belief around the middle of the 19th century that many concepts in algebra must have an important meaning for differential equations if they were appropriately generalized. Following these ideas Cockle [31] introduced a notion that came fairly close to an invariant of a differential equation in terms of his so-called criticoids. He obtained them for linear second order ode’s by elementary methods without applying the notion of a group of transformations. Laguerre [98] was the first to realize the close connection between the type of transformations admitted by a linear second order equation and the invariants belonging to it. Brioschi [19] and Halphen [65] generalized these results to equations of third and fourth order. For quasilinear equations of first and second order Liouville [117, 120] determined various classes of invariants and distinguished absolute and relative invariants. Later, Tresse [181] in his Preisschrift presented an extensive discussion of the invariants of the equation y′′ = F (x, y, y′) and solved the corresponding equivalence problem. Some of his results have been generalized to third order equations of the form y′′′ = F (x, y, y′, y′′) by Leja [103]. A survey on the history of the subject, including some comments and reprints of relevant articles may be found in the booklet by Czichowski and Fritzsche [36]. A more modern treatement of the subject is given in the book by Olver [141].
