Title: On C-class equations,
Abstract: The concept of a C-class of differential equations goes back to E. Cartan
with the upshot that generic equations in a C-class can be solved without
integration. While Cartan's definition was in terms of differential invariants
being first integrals, all results exhibiting C-classes that we are aware of
are based on the fact that a canonical Cartan geometry associated to the
equations in the class descends to the space of solutions. For sufficiently low
orders, these geometries belong to the class of parabolic geometries and the
results follow from the general characterization of geometries descending to a
twistor space.
In this article we answer the question of whether a canonical Cartan geometry
descends to the space of solutions in the case of scalar ODEs of order at least
four and of systems of ODEs of order at least three. As in the lower order
cases, this is characterized by the vanishing of the generalized Wilczynski
invariants, which are defined via the linearization at a solution. The
canonical Cartan geometries (which are not parabolic geometries) are a slight
variation of those available in the literature based on a recent general
construction. All the verifications needed to apply this construction for the
classes of ODEs we study are carried out in the article, which thus also
provides a complete alternative proof for the existence of canonical Cartan
connections associated to higher order (systems of) ODEs.