On a question of Gundersen-Yang concerning entire solutions of binomial differential equations
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| Format: | Preprint |
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2025
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| _version_ | 1866915105346682880 |
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| author | Long, Jianren Xia, Mengting Xiang, Xuxu |
| author_facet | Long, Jianren Xia, Mengting Xiang, Xuxu |
| contents | We study the question posed by G. Gundersen and C. C. Yang, in which the following two types of binomial differential equations are investigated, $$ a(z)f'f''-b(z)(f)^{2}=c(z)e^{2d(z)},~~a(z)ff'-b(z)(f'')^{2}=c(z)e^{2d(z)}, $$ where $a(z)$, $b(z)$ and $c(z)$ are polynomials such that $a(z)b(z)c(z)\not\equiv 0$, $d(z)$ is non-constant polynomial. The explicit forms of entire solutions of the above binomial differential equations are obtained by using the Nevanlinna theory, which gives partial solutions to the question of G. Gundersen and C. C. Yang. In addition, some examples are given to illustrate these results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a question of Gundersen-Yang concerning entire solutions of binomial differential equations Long, Jianren Xia, Mengting Xiang, Xuxu Complex Variables 34M10, 34M05, 30D35 We study the question posed by G. Gundersen and C. C. Yang, in which the following two types of binomial differential equations are investigated, $$ a(z)f'f''-b(z)(f)^{2}=c(z)e^{2d(z)},~~a(z)ff'-b(z)(f'')^{2}=c(z)e^{2d(z)}, $$ where $a(z)$, $b(z)$ and $c(z)$ are polynomials such that $a(z)b(z)c(z)\not\equiv 0$, $d(z)$ is non-constant polynomial. The explicit forms of entire solutions of the above binomial differential equations are obtained by using the Nevanlinna theory, which gives partial solutions to the question of G. Gundersen and C. C. Yang. In addition, some examples are given to illustrate these results. |
| title | On a question of Gundersen-Yang concerning entire solutions of binomial differential equations |
| topic | Complex Variables 34M10, 34M05, 30D35 |
| url | https://arxiv.org/abs/2501.09250 |