Meromorphic Solutions of a Certain Type of Nonlinear Differential Equations

In this paper, using Nevanlinna theory and linear algebra, we characterize transcendental meromorphic solutions of nonlinear differential equation of the form \(\begin{aligned} f^n+Q_d(z,f)=\sum _{i=1}^{l}p_{i}(z)e^{\alpha _{i}(z)}, \end{aligned}\) $$\begin{array}{c}\hfill f^n+Q_d(z,f)=\sum _{i=1}^l{p}_i(z)e^{{\alpha }_i(z)},\end{array}$$ where \(l\ge 2\) $l\ge 2$ , \(n\ge l+2\) $n\ge l+2$ are integers, f(z) is a meromorphic function, \(Q_d(z,f)\) $Q_d(z,f)$ is a differential polynomial in f(z) of degree \(d\le n-(l+1)\) $d\le n-(l+1)$ with rational functions as its coefficients, \(p_{1}(z)\) $p_1(z)$ , \(p_{2}(z)\) $p_2(z)$ , \(\dots \) $\cdots$ , \(p_{l}(z)\) $p_l(z)$ are non-vanishing rational functions and \(\alpha _{1}(z)\) ${\alpha }_1(z)$ , \(\alpha _{2}(z)\) ${\alpha }_2(z)$ , \(\dots \) $\cdots$ , \(\alpha _{l}(z)\) ${\alpha }_l(z)$ are nonconstant polynomials such that \(\alpha _{1}^\prime (z)\) ${\alpha }_1^{\prime }(z)$ , \(\alpha _{2}^\prime (z)\) ${\alpha }_2^{\prime }(z)$ , \(\dots \) $\cdots$ , \(\alpha _{l}^\prime (z)\) ${\alpha }_l^{\prime }(z)$ are distinct. Further, we give the necessary conditions for the existence of meromorphic solutions of the above equation, and supply the example to demonstrate the sharpness of the condition of the obtained theorem.