On transcendental meromorphic solutions of certain types of differential equations

TL;DR

This paper investigates the solutions of a new class of nonlinear differential equations involving transcendental meromorphic functions, correcting previous results and providing explicit solution forms with illustrative examples.

Contribution

It introduces and analyzes a novel type of nonlinear differential equation involving meromorphic functions, correcting prior inaccuracies and characterizing solution structures.

Findings

Corrected a lacuna in previous results by Xue

Determined explicit solution forms for the new differential equations

Provided examples validating the theoretical results

Abstract

In this paper, for a transcendental meromorphic function $f$ and $a\in \mathbb{C}$, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: \beas f^n+af^{n-2}f'+ P_d(z,f) = \sum_{i=1}^{k}p_i(z)e^{\alpha_i(z)},\eeas where $P_d(z,f)$ is differential polynomial of $f$, $p_i$'s and $\alpha_{i}$'s are non-vanishing rational functions and non-constant polynomials respectively. When $a=0$, we have pointed out a major lacuna in a recent result of Xue [Math. Slovaca, 70(1)(2020), 87-94] and rectifying the result, presented the corrected form of the same at a large extent. The case $a\neq 0$ has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.