Meromorphic solutions of a certain type of nonlinear differential equation
Authors
Sujoy majumder
Nabadwip Sarkar
Debabrata pramanik
Abstract
Our paper focuses on investigating the existence and possible forms of solutions to the nonlinear differential equation \beas f^m+\big(Rf^{(k)}\big)^n=Qe^α,\eeas where where $k$, $m$ and $n$ are three positive integers, $Q$ and $R$ are non-zero rational functions, and $α$ is a polynomial. We examine this equation systematically for all positive integral values of $k$, $m$ and $n$, providing comprehensive conditions under which entire or meromorphic solutions may exist. Our results offer substantial improvements over those of Tang and Liao \cite{TL1} and Han and Lü \cite{HL1}, particularly with respect to the existence criteria and the explicit analytic forms of admissible solutions. These findings contribute meaningfully to the broader study of nonlinear differential polynomials and Fermat-type functional equations in complex analysis.
