Meromorphic solution of a certain type of algebraic differential equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911151307096064 |
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| author | Xu, Junfeng Majumder, Sujoy Sarkar, Nabadwip Mahato, Lata |
| author_facet | Xu, Junfeng Majumder, Sujoy Sarkar, Nabadwip Mahato, Lata |
| contents | In the paper, we use the idea of normal family to find out the possible solution of the following special case of algebraic differential equation \[P_k\big(z,f,f^{(1)},\ldots, f^{(k)}\big)=f^{(1)}(f-\mathscr{L}_k(f))-φ(f-a)(f-b)=0,\] where $\mathscr{L}_k(f)=\sideset{}{_{i=0}^k}{\sum} a_i f^{(i)}$ and $φ$ is an entire function, $a_i\in\mathbb{C}\;(i=0,1,\ldots, k)$ such that $a_k=1$ and $a, b\in\mathbb{C}$ such that $a\neq b$. The obtained results improve and generalise the results of Li and Yang \cite{LY1} and Xu et al. \cite{XMD} in a large scale. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Meromorphic solution of a certain type of algebraic differential equation Xu, Junfeng Majumder, Sujoy Sarkar, Nabadwip Mahato, Lata Complex Variables 30D35, 30D45 and 34M10 In the paper, we use the idea of normal family to find out the possible solution of the following special case of algebraic differential equation \[P_k\big(z,f,f^{(1)},\ldots, f^{(k)}\big)=f^{(1)}(f-\mathscr{L}_k(f))-φ(f-a)(f-b)=0,\] where $\mathscr{L}_k(f)=\sideset{}{_{i=0}^k}{\sum} a_i f^{(i)}$ and $φ$ is an entire function, $a_i\in\mathbb{C}\;(i=0,1,\ldots, k)$ such that $a_k=1$ and $a, b\in\mathbb{C}$ such that $a\neq b$. The obtained results improve and generalise the results of Li and Yang \cite{LY1} and Xu et al. \cite{XMD} in a large scale. |
| title | Meromorphic solution of a certain type of algebraic differential equation |
| topic | Complex Variables 30D35, 30D45 and 34M10 |
| url | https://arxiv.org/abs/2509.10113 |