关键词: Nevanlinna theory, nonlinear differential equations, meromorphic functions, entire functions

Abstract: We find the exact forms of meromorphic solutions of the nonlinear differential equations $f^n+q(z){\rm e}^{Q(z)}f^{(k)}=p_1{\rm e}^{\alpha_1 z}+p_2{\rm e}^{\alpha_2 z}, \quad n\geq3, ~k\geq1,$ where $q, Q$ are nonzero polynomials, $Q\not\equiv Const.$, and $p_1, p_2, \alpha_1, \alpha_2$ are nonzero constants with $\alpha_1\neq\alpha_2$. Compared with previous results on the equation $p(z)f^3+q(z)f''=-\sin \alpha(z)$ with polynomial coefficients, our results show that the coefficient of the term $f^{(k)}$ perturbed by multiplying an exponential function will affect the structure of its solutions.

Key words: Nevanlinna theory, nonlinear differential equations, meromorphic functions, entire functions