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

TL;DR

This paper investigates the differential invariants of the general second-order ODEs under point transformations, extending previous work to non-cubic cases and providing a comprehensive invariant classification.

Contribution

It derives all fifth order differential invariants and invariant differentiation operators for the general class of second-order ODEs, expanding the understanding of their invariance properties.

Findings

Determined all fifth order differential invariants.

Provided invariant descriptions of canonical forms for non-zero Tresse invariants.

Extended the invariance analysis beyond cubic nonlinear cases.

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.