Painleve Classification of Polynomial Ordinary Differential Equations of Arbitrary Order and Second Degree

TL;DR

This paper provides a complete Painleve classification for polynomial ODEs of arbitrary order and second degree, identifying seven classes with specific integrability and linearizability properties, and introduces a new necessary condition for Painleve property.

Contribution

It introduces a comprehensive classification of polynomial ODEs of arbitrary order with second degree, including a novel necessary condition for Painleve property applicable to broader cases.

Findings

Seven classes of equations with Painleve property identified

Five classes of order up to four are previously known

One class of arbitrary order is linearizable

Abstract

The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work we propose the complete Painleve classification for ordinary differential equations of the arbitrary order with right-hand side being a quadratic form on the dependent variable and all of its derivatives. The total of seven classes of the equations with Painleve property have been found. Five of them having the order up to four are already known. Sixth one of the other up to five also appears to be integrable in the known functions. While the only seventh class of the unrestricted order appears to be linearizable. The classification employs a novel general necessary condition for the Painleve property proven in the paper, potentially having a broader application for the Painleve classification of other types of ordinary differential equations.