On meromorphic solutions of non-linear differential equations of Tumura-Clunie type

TL;DR

This paper studies meromorphic solutions of certain nonlinear differential equations of Tumura-Clunie type, focusing on cases where the solutions are linked to linear differential equations with rational coefficients, extending previous results.

Contribution

It generalizes existing results by analyzing solutions where the function h satisfies a linear differential equation with rational coefficients, broadening the class of equations studied.

Findings

Solutions h have rational order under specified conditions.

Extends previous results to more general forms of h.

Provides new insights into the structure of meromorphic solutions.

Abstract

Meromorphic solutions of non-linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\geq 2$ is an integer, $h$ is a meromorphic function, and $P(z,f)$ is differential polynomial in $f$ and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when $h$ has the particular form $h(z)=p_1(z)e^{\alpha_1(z)}+p_2(z)e^{\alpha_2(z)}$, where $p_1, p_2$ are small functions of $f$ and $\alpha_1,\alpha_2$ are entire functions. In such a case the order of $h$ is either a positive integer or equal to infinity. In this article it is assumed that $h$ is a meromorphic solution of the linear differential equation $h'' +r_1(z) h' +r_0(z) h=r_2(z)$ with rational coefficients $r_0,r_1,r_2$, and hence the order of $h$ is a rational number. Recent results by Liao-Yang-Zhang (2013) and Liao (2015) follow as special cases of the main results.