Non-linear differential equations and Hayman’s theorem on differential polynomials

Abstract

In this paper, we first prove that if the following differential equationadmits a meromorphic function with finitely many poles, where and is a differential polynomial in with degree and rational functions as its coefficients, is a non-zero rational function and is a non-constant polynomial, then has the form and where is a rational function and is a polynomial with With this in hand, we prove if is a transcendental entire function, is a polynomial of degree , then assumes every complex number infinitely many times, except a possible value . On the other hand, if assumes the complex value finitely many times, then and .