Bulletin of the
Korean Mathematical Society
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Bull. Korean Math. Soc.

Published online July 5, 2022

Copyright © The Korean Mathematical Society.

On transcendental meromorphic solutions of certain types of differential equations

Abhijit Banerjee, Tania Biswas, and Sayantan Maity

University of Kalyani

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. In addition, our main result is also an improvement of a recent result of Chen-Lian [Bull. Korean Math. Soc., 54(4)(2020), 1061-1073] by rectifying a gap in the proof of the theorem of the same paper. 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.

Keywords: Non-linear differential equations, meromorphic solutions, Nevanlinna theory, exponential

MSC numbers: 34A34, 34M05, 30D35

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