Second order ODE's cubic in the first order derivative with 2-dimensional symmetry algebra

TL;DR

This paper classifies second order ODEs cubic in the first derivative with 2-dimensional symmetry algebra and provides an equivalence criterion similar to Lie's linearization test.

Contribution

It identifies eight distinct types of such ODEs and develops a straightforward equivalence criterion for their classification.

Findings

Eight types of second order ODEs cubic in the first derivative identified

An easily verifiable equivalence criterion established for each type

Analogy to Lie's linearization criterion demonstrated

Abstract

We describe the second order ODE's cubic in the first order derivative with 2-dimensional symmetry algebra. We show that there exist only eight different types of them. We also construct the easily verifiable Equivalence Criterion for every type of equations analogous to Linearization Criterion of S. Lie.