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 \( \gamma_P \leq n - 1 \) 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.