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} $

Abstract

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.

Keywords

• nevanlinna theory
• complex differential equations
• exponential sums
• meromorphic solutions