On the Property of Solutions for a Class of Higher Order Periodic Differential Equations

DOI：10.3770/j.issn:2095-2651.2013.04.005

 作者 单位 肖丽鹏 江西师范大学数学与信息科学学院, 江西 南昌 330022

本文研究了高阶线性微分方程$$f^{(k)}(z)+A_{k-2}(z)f^{(k-2)}(z)+\cdots+A_0(z)f(z)=0,\eqno(*)$$解的线性相关性,其中$A_j(z)(j=0,2,\ldots,k-2)$是常数, $A_1$为非常数的的整周期函数,周期为$2\pi i$,且是$e^z$的有理函数.在一定条件下,我们给出了方程(*)解的表示.

In this paper, the property of linear dependence of solutions for higher order linear differential equation $$f^{(k)}(z)+A_{k-2}(z)f^{(k-2)}(z)+\cdots+A_0(z)f(z)=0,\eqno(*)$$ where $A_j(z)~(j=0,2,\ldots,k-2)$ are constants and $A_1$ is a non-constant entire function of period $2\pi i$ and rational in $e^z$, is investigated. Under certain condition, the representation of solution of Eq.\,$(*)$ is given, too.