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arXiv 2103.14893math.CV
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一类微分方程的超越亚纯解

On transcendental meromorphic solutions of certain types of differential equations

Tania Biswas, Sayantan Maity, Abhijit Banerjee

摘要:

本文对超越亚纯函数$f$和$a\in \mathbb{C}$,详尽地研究了一类新的非线性微分方程解的性质和形式,该方程具有以下形式,这是以前从未研究过的:\beas f ^n + af ^{n-2} f ′ + P_d(z,f)= \sum_{i=1}^{k} p_i(z)e ^{α_i(z)},\eeas其中$P_d(z,f)$是$f$的微分多项式,$p_i$和$α_{i}$分别是非零有理函数和非常数多项式。当$a=0$时,我们指出了薛[Math.Slovaca,70(1)(2020),87 - 94]最近结果中的一个主要缺陷,并纠正了该结果,在很大程度上给出了该结果的校正形式。还对病例$a\neq 0$进行了操作,以确定溶液的形式。我们还举例说明了一些例子,以显示我们的结果的准确性。

英文摘要:

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^{α_i(z)},\eeas where $P_d(z,f)$ is differential polynomial of $f$, $p_i$'s and $α_{i}$'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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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