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.