Meromorphic Solutions of Certain Types of Non-linear Differential Equations

Abstract

Let $$p_1, p_2$$ and $$\alpha _1, \alpha _2$$ be non-zero constants, and $$P_d(z, f)$$ be a differential polynomial in f of degree d. Li obtained the forms of meromorphic solutions with few poles of the non-linear differential equations $$f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}$$ provided $$\alpha _1\ne \alpha _2$$ and $$d\le n-2$$. In this paper, given $$d=n-1$$, we find the forms of meromorphic solutions with few poles of the above equations under some restrictions on $$\alpha _1, \alpha _2$$. Some examples are given to illustrate our results.