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

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 \( γ_P \leq n - 1 \) 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.

13 pages