Group Classification of a family of second-order differential equations

TL;DR

This paper determines the symmetry groups of a class of second-order differential equations and provides a complete classification based on these symmetries, aiding in their analysis and solution.

Contribution

It introduces a complete group classification for equations of the form y''=A(x)y'+F(y) using a direct analysis method and equivalence transformations.

Findings

Identified the symmetry group for the differential equations.

Classified equations based on their symmetry properties.

Provided a systematic framework for analyzing similar differential equations.

Abstract

We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method of analysis, together with the equivalence transformations.