# Differential invariants and equivalence of ODEs $y''=a^3(x,y)y'^3+a^2(x,y)y'^2+a^1(x,y)y'+a^0(x,y)$

## TL;DR

This paper develops the algebra of differential invariants for a class of third-order polynomial ODEs and solves their equivalence problem under point transformations, advancing classification methods in differential geometry.

## Contribution

It constructs the algebra of differential invariants and solves the equivalence problem for a specific class of third-order polynomial ODEs.

## Key findings

- Algebra of differential invariants is explicitly constructed.
- Equivalence problem for the class of equations is solved.
- Provides a classification framework for these ODEs.

## Abstract

This paper is devoted to ordinary differential equations of the form $$y''=a^3(x,y)y'^3+a^2(x,y)y'^2+a^1(x,y)y'+a^0(x,y)$$ The algebra of all differential invariants of point transformations is constructed for these equations in general position and the equivalence problem is solved for them.