Meromorphic solutions of two certain types of nonlinear differential equations

Abstract

We investigate the meromorphic solutions of two types of nonlinear differential equations of the form

$$\begin{array}{lllllll}\hfill b{f}^n+a{f}^{n-1}{f}^{\prime }+Q_d\left(f\right)& =u\left(z\right)e^{v\left(z\right)},& \hfill n\ge d+2,& & \hfill & & \hfill \\ \hfill f^n+f^{n-1}{f}^{\prime }+Q_d\left(f\right)& =P_1\left(z\right)e^{{\alpha }_1}+P_2\left(z\right)e^{{\alpha }_2},& \hfill n\ge d+3,& & \hfill & & \hfill \end{array}$$

where $a,b$ are constants with $\left(a,b\right)\ne \left(0,0\right)$ and $n,d$ are positive integers, $u,P_1,P_2$ are nonzero rational functions, $v,{\alpha }_1,{\alpha }_2$ are nonconstant polynomials, and $Q_d\left(f\right)$ denotes a differential polynomial in $f$ with rational functions as its coefficients. Our results improve some recent related results.