On the contact equivalence problem of second order ODEs which are quadratic with respect to the second order derivative

TL;DR

This paper derives necessary and sufficient conditions for contact equivalence of quadratic second-order ODEs, using integrals of specific distributions, advancing the understanding of differential equation transformations.

Contribution

It provides a complete characterization of contact equivalence for quadratic second-order ODEs in terms of integrals, a novel approach in the field.

Findings

Established criteria for contact equivalence of quadratic second-order ODEs.

Connected contact transformations to integrals of one-dimensional distributions.

Enhanced methods for classifying second-order differential equations.

Abstract

In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form $y"{}^2+A(x,y,y') y"+B(x,y,y')=0$ to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional distributions.