Point Equivalence of Second-Order ODEs: Maximal Invariant Classification Order

Abstract

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlev\'e-I equation can be characterized as the simplest second-order ODE belonging to the class of equations requiring 10th order jets for their classification.