Tumura-Clunie Differential Equations with Applications to Linear ODE's

TL;DR

This paper investigates nonlinear Tumura-Clunie type differential equations with meromorphic solutions, classifies solutions based on zero and pole distribution, and applies findings to entire solutions of higher-order linear differential equations.

Contribution

It introduces a classification of solutions to Tumura-Clunie differential equations when the inhomogeneous term satisfies a linear differential equation, extending existing results.

Findings

Solutions are classified into two cases based on zeros and poles.

Results apply to zeros and critical points of entire solutions.

Extends known results in the theory of differential equations.

Abstract

In this paper, we study nonlinear differential equations of Tumura-Clunie type, $ f^n + P(z, f) = h, $ where \( n \geq 2 \) is an integer, \( P(z, f) \) is a differential polynomial in \( f \) of degree \( \gamma_P \leq n - 1 \) with small functions as coefficients, and \( h \) is a meromorphic function. Assuming that $ h $ satisfies a linear differential equation of order $ p\le n $ with rational coefficients, we establish a result that classifies the meromorphic solutions \( f \) into two cases based on the distribution of their zeros and poles. This result is then applied to study the zeros and critical points of entire solutions to certain higher-order linear differential equations, thereby extending some known results in the literature.