On invariants of second-order ordinary differential equations $y''=f(x,y,y')$ via point transformations

Abstract

Bagderina \cite{Bagderina2013} solved the equivalence problem for a family of scalar second-order ordinary differential equations (ODEs), with cubic nonlinearity in the first-order derivative, via point transformations. However, the question is open for the general class $y''=f(x,y,y')$ which is not cubic in the first-order derivative. We utilize Lie's infinitesimal method to study the differential invariants of this general class under an arbitrary point equivalence transformations. All fifth order differential invariants and the invariant differentiation operators are determined. As an application, invariant description of all the canonical forms in the complex plane for second-order ODEs $y''=f(x,y,y')$ where both of the two Tress\'e relative invariants are non-zero is provided.