On the Non-existence of Meromorphic Solutions of Certain Types of Non-linear Differential Equations

Abstract

In this paper, we study meromorphic solutions of non-linear differential equations of the form $$f^n+P_d(f)=p_1e^{\alpha _1(z)}+p_2e^{\alpha _2(z)}$$ , where $$\alpha _1,\alpha _2$$ are polynomials of degree $$k(\ge 1)$$ , $$p_1$$ , $$p_2$$ are small meromorphic functions of $$e^{z^k}$$ , $$P_\mathrm{d}(f)$$ is a differential polynomial in f of degree d with small meromorphic functions of f as its coefficients. Some sufficient conditions on the non-existence of meromorphic solutions of such equations are provided. Our results complement some previous results.