关于一类常微分方程亚纯解的个数的注记

A NOTE OF ON THE NUMBER OF THE MEROMORPHIC SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS

本文讨论微分方程u′=sum from k=0 P_k(z)u^k (1)在某些条件下亚纯解的个数问题。得到定理1、2,它们分别是[1]定理1、2的推广。

In this paper we obtain the following theorems Theorem 1; for n≥3, the differential eqationwhere each P_k is ameromorphic function possessing a finite number of poles and R(z) is ra-tional, g(z) is entire can possess at most a finite number of distinct meromorphic solutions. Themorem 2; For n≥3, the differential equationwhere P_k(z);(k=0,...,n) are meromorphic functions such that the poles of each P_k(z)have exponent of convergence less that one and the zeros of P_k(z) have exponent ofconvergence less than one and the zeros of P_k(z) have exponent of convergence less thanone. If there are two meromorphic solutions u_1,u_2 of Eq.(2) such that p((u_2-u_1)P_n)≥1,then Eq.(2) possesses at most then number of distinct meromorphic solutions.