Hayman's classical conjecture on some nonlinear second order algebraic ODEs

TL;DR

This paper investigates the growth of meromorphic solutions to certain nonlinear second order algebraic ODEs, explicitly finds all solutions, and confirms Hayman's classical conjecture for these equations.

Contribution

It provides explicit solutions and verifies Hayman's conjecture for specific classes of nonlinear second order algebraic ODEs.

Findings

All solutions are meromorphic and explicitly determined.

The studied ODEs satisfy Hayman's classical conjecture.

Growth of solutions is characterized in terms of the Nevanlinna characteristic.

Abstract

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second order nonlinear algebraic ordinary differential equations. We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the {\it classical conjecture} proposed by Hayman in 1996.