On the point transformations for the second order differential equations. I

TL;DR

This paper investigates point transformations for a class of second-order differential equations, revisiting classical results and addressing the equivalence problem for equations in general position.

Contribution

It provides a comprehensive review of classical results and offers a solution to the equivalence problem for these differential equations.

Findings

Classical results on point transformations are summarized.

A solution to the equivalence problem for equations in general position is presented.

The paper clarifies the structure of transformations for the specified differential equations.

Abstract

Point transformations for the ordinary differential equations of the form $y''=P(x,y)+3 Q(x,y) y'+3 R(x,y) (y')^2+S(x,y) (y')^3$ are considered. Some classical results are resumed. Solution for the equivalence problem for the equations of general position is described.