Meromorphic solutions of higher order Briot-Bouquet differential equations

TL;DR

This paper proves that meromorphic solutions with poles of higher order Briot-Bouquet differential equations are elliptic functions, extending the understanding of their solution structure.

Contribution

It establishes that all meromorphic solutions with poles for these equations are elliptic functions, including degenerate cases, providing a comprehensive classification.

Findings

Meromorphic solutions with poles are elliptic functions.

Degenerate elliptic functions are also solutions.

The result generalizes previous classifications of such differential equations.

Abstract

For differential equations $P(y^{(k)},y)=0,$ where $P$ is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.