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

In this paper, we study nonlinear differential equations of Tumura-Clunie type, $f^n+P(z,f)=h,$ where $n\ge 2$ is an integer, $P(z,f)$ is a differential polynomial in $f$ of degree ${\gamma }_P\le n-1$ 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.