某类二阶线性微分方程的解的增长性  

On the Growth of Solutions of a Class of Second Order Linear Differential Equations

作　　者：吴昕[1] 肖丽鹏[1] 

机构地区：[1]江西师范大学数学与信息科学学院,南昌330022

出　　处：《工程数学学报》2015年第6期898-908,共11页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11301232;11171119);江西省自然科学基金(20132BAB211009);江西省教育厅青年科学基金(GJJ12207)~~

摘　　要：本文主要研究某类二阶线性微分方程解的增长性.这类方程的系数是关于复指数函数的多项式,且多项式的系数又是超越整函数.我们论证指出:当这类方程的系数满足一定条件时,方程的每一个非平凡解的超级必为1.我们利用值分布的相关理论,分两步进行证明:第一步,利用反证法和超越亚纯函数的性质,证明所考虑方程的每一个非平凡解的增长级必为无穷;第二步,利用反证法及Wiman-Valiron理论,证明方程的每一个非平凡解的超级为1.本文得到的结果完善了前人相关结果.The aim of this paper is to consider the growth of solutions of certain second-order linear differential equation. The coefficients of the equation are polynomials in the complex exponential function, while the coefficients of the polynomials are transcendental integral functions. The value distribution theory is mainly used to show that the hyper-order of every nontrivial solution of the equation equals one when the coefficients of the equation satisfy certain conditions. The proof can be divided into two steps： firstly, it is shown that the growth order of every nontrivial solution of the considered equation equals infinity by contradiction and the properties of transcendental meromorphic functions; secondly, it is shown that the hyper-order of every nontrivial solution of the equation equals one by contradiction and the Wiman-Valiron theory. The obtained results generalize some previous results.

关 键 词：线性微分方程 超越整函数 超级 解的增长级 

分 类 号：O174.52[理学—数学] O175.12[理学—基础数学]