Meromorphic solutions of non‐linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\ge 2$ is an integer, h is a meromorphic function, and $P(z,f)$ is differential polynomial in f and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when h has the particular form $h(z)=p_1(z)e^{{\alpha }_1(z)}+p_2(z)e^{{\alpha }_2(z)}$, where $p_1,p_2$ are small functions of f and ${\alpha }_1,{\alpha }_2$ are entire functions. In such a case the order of h is either a positive integer or equal to infinity. In this article it is assumed that h is a meromorphic solution of the linear differential equation $h^{\prime \prime }+r_1(z)h^{\prime }+r_0(z)h=r_2(z)$ with rational coefficients $r_0,r_1,r_2$, and hence the order of h is a rational number. Recent results by Liao–Yang–Zhang (2013) and Liao (2015) follow as special cases of the main results.

中文翻译：

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Tumura–Clunie型非线性微分方程的亚纯解

形式为非线性微分方程的亚纯解 $F^{ñ}+P（ž，F）=H$ 被调查，在哪里 $ñ\ge \mathrm{2个}$是整数，h是亚纯函数，并且$P（ž，F）$是f及其微分函数的微分多项式，其系数很小。在现有文献中，当h具有特定形式时已经研究了该方程$H（ž）=p_{\mathrm{1个}}（ž）{Ë}^{{\alpha }_{\mathrm{1个}}（ž）}+p_{\mathrm{2个}}（ž）{Ë}^{{\alpha }_{\mathrm{2个}}（ž）}$， 在哪里 $p_{\mathrm{1个}}，p_{\mathrm{2个}}$是f和的小函数${\alpha }_{\mathrm{1个}}，{\alpha }_{\mathrm{2个}}$是全部功能。在这种情况下，h的阶为正整数或等于无穷大。在本文中，假设h是线性微分方程的亚纯解$H^{\prime \prime }+{\mathrm{[R}}_{\mathrm{1个}}（ž）H^{\prime }+{\mathrm{[R}}_0（ž）H={\mathrm{[R}}_{\mathrm{2个}}（ž）$ 具有合理的系数 ${\mathrm{[R}}_0，{\mathrm{[R}}_{\mathrm{1个}}，{\mathrm{[R}}_{\mathrm{2个}}$，因此h的阶为有理数。廖扬章（2013）和廖扬（2015）的最新研究结果是主要结果的特例。