MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS

doi:10.1007/s10473-025-0321-5

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 1155-1168.doi: 10.1007/s10473-025-0321-5

MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS

Minfeng CHEN^1,†, Meili LIANG¹, Zongsheng GAO²

1. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China;
2. LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China

收稿日期:2023-12-26 修回日期:2024-04-23 出版日期:2025-05-25 发布日期:2025-09-30

MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS

Minfeng CHEN^1,†, Meili LIANG¹, Zongsheng GAO²

1. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China;
2. LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China

Received:2023-12-26 Revised:2024-04-23 Online:2025-05-25 Published:2025-09-30
Contact: ^†Minfeng CHEN, E-mail: chenminfeng198710@126.com
About author:Meili LIANG, E-mail: liangml@gdufs.edu.cn; Zongsheng GAO, E-mail: zshgao@buaa.edu.cn
Supported by:
National Natural Science Foundation of China (No. 12001117), and the Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515110654).

摘要/Abstract

摘要： In this article,~we study the meromorphic solutions of the following non-linear differential equation $\begin{equation*} f^{n}(z)+P_{d}(z,f)=\sum_{j=1}^{k}p_{j}{\rm e}^{\alpha_{j}z}, \end{equation*}$ where $n$ and $k$ are integers with $n\geq k\geq 3$, $P_{d}(z,f)$ is a differential polynomial in $f$ of degree $d\leq n-1$, $p_{j}'s$ and $\alpha_{j}'s$ are non-zero constants. We obtain the expressions of meromorphic solutions of the above equations under some restrictions on $\alpha_{j}'s$. Some examples are given to illustrate the possibilities of our results.

关键词: meromorphic solution, non-linear differential equation, Nevanlinna theory, exponential, differential polynomial

Abstract: In this article,~we study the meromorphic solutions of the following non-linear differential equation $\begin{equation*} f^{n}(z)+P_{d}(z,f)=\sum_{j=1}^{k}p_{j}{\rm e}^{\alpha_{j}z}, \end{equation*}$ where $n$ and $k$ are integers with $n\geq k\geq 3$, $P_{d}(z,f)$ is a differential polynomial in $f$ of degree $d\leq n-1$, $p_{j}'s$ and $\alpha_{j}'s$ are non-zero constants. We obtain the expressions of meromorphic solutions of the above equations under some restrictions on $\alpha_{j}'s$. Some examples are given to illustrate the possibilities of our results.

Key words: meromorphic solution, non-linear differential equation, Nevanlinna theory, exponential, differential polynomial

中图分类号:

34M05

Minfeng CHEN, Meili LIANG, Zongsheng GAO. MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS[J]. 数学物理学报(英文版), 2025, 45(3): 1155-1168.

Minfeng CHEN, Meili LIANG, Zongsheng GAO. MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS[J]. Acta mathematica scientia,Series B, 2025, 45(3): 1155-1168.

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MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS

MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS WITH $K$ EXPONENTIAL TERMS

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