Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $
By using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equation $ \begin{equation*} f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z}, \end{equation*} $ where $...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-08-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20221007?viewType=HTML |
| _version_ | 1817998656834371584 |
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| author | Linkui Gao Junyang Gao |
| author_facet | Linkui Gao Junyang Gao |
| author_sort | Linkui Gao |
| collection | DOAJ |
| description | By using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equation
$ \begin{equation*} f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z}, \end{equation*} $
where $ P_{d}(f) $ is a differential polynomial in $ f $ of degree $ d(0\leq d\leq n-3) $ with small meromorphic coefficients and $ p_{i}, \alpha_{i}(i = 1, 2, 3) $ are nonzero constants. We show that the solutions of this type equation are exponential sums and they are in $ \Gamma_{0}\cup\Gamma_{1}\cup\Gamma_{3} $ which will be given in Section $ 1 $. Moreover, we give some examples to illustrate our results. |
| first_indexed | 2024-04-14T02:55:54Z |
| format | Article |
| id | doaj.art-8490b8932dad467d9e0f2ed118af6fd8 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-14T02:55:54Z |
| publishDate | 2022-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-8490b8932dad467d9e0f2ed118af6fd82022-12-22T02:16:06ZengAIMS PressAIMS Mathematics2473-69882022-08-01710182971831010.3934/math.20221007Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $Linkui Gao0Junyang Gao1School of Science, China University of Mining and Technology, Beijing 100083, ChinaSchool of Science, China University of Mining and Technology, Beijing 100083, ChinaBy using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equation $ \begin{equation*} f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z}, \end{equation*} $ where $ P_{d}(f) $ is a differential polynomial in $ f $ of degree $ d(0\leq d\leq n-3) $ with small meromorphic coefficients and $ p_{i}, \alpha_{i}(i = 1, 2, 3) $ are nonzero constants. We show that the solutions of this type equation are exponential sums and they are in $ \Gamma_{0}\cup\Gamma_{1}\cup\Gamma_{3} $ which will be given in Section $ 1 $. Moreover, we give some examples to illustrate our results.https://www.aimspress.com/article/doi/10.3934/math.20221007?viewType=HTMLnevanlinna theorycomplex differential equationsexponential sumsmeromorphic solutions |
| spellingShingle | Linkui Gao Junyang Gao Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $ AIMS Mathematics nevanlinna theory complex differential equations exponential sums meromorphic solutions |
| title | Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $ |
| title_full | Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $ |
| title_fullStr | Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $ |
| title_full_unstemmed | Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $ |
| title_short | Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $ |
| title_sort | meromorphic solutions of f n p d f p 1 e alpha 1 z p 2 e alpha 2 z p 3 e alpha 3 z |
| topic | nevanlinna theory complex differential equations exponential sums meromorphic solutions |
| url | https://www.aimspress.com/article/doi/10.3934/math.20221007?viewType=HTML |
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