Bull. Korean Math. Soc. 2022; 59(5): 1145-1166
Online first article July 5, 2022 Printed September 30, 2022
https://doi.org/10.4134/BKMS.b210637
Copyright © The Korean Mathematical Society.
Abhijit Banerjee, Tania Biswas, Sayantan Maity
University of Kalyani; University of Kalyani; University of Kalyani
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: \[f^n+af^{n-2}f'+ P_d(z,f) = \sum_{i=1}^{k}p_i(z)e^{\alpha_i(z)}, \] where $P_d(z,f)$ is a 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 [17] and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian [2] 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
Supported by: The authors wish to thank the referee for thorough review. Abhijit Banerjee is thankful to DST-PURSE II programme for financial assistance. Tania Biswas is thankful to the University Grant Commission (UGC), Govt. of India for financial support under UGC-Ref. No.: 1174/(CSIR-UGC NET DEC. 2017) dated 21/01/2019. Sayantan Maity wish to thank Council of Scienti c and Industrial Research (CSIR, India) for providing fellowship under File No: 09/106(0191)/2019-EMR-I.
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