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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 898-908.doi: 10.3969/j.issn.1005-3085.2015.06.011

• • 上一篇    下一篇

某类二阶线性微分方程的解的增长性

吴  昕,   肖丽鹏   

  1. 江西师范大学数学与信息科学学院，南昌 330022
  • 收稿日期:2014-06-06 接受日期:2015-04-09 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    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).

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

WU Xin,   XIAO Li-peng   

  1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022
  • Received:2014-06-06 Accepted:2015-04-09 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The National Natural Science Foundation of China (11301232; 11171119); the Natural Science Foundation of Jiangxi Province (20132BAB211009); the Youth Science Foundation of Education Bureau of Jiangxi Province (GJJ12207).

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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．

关键词: 线性微分方程, 超越整函数, 超级, 解的增长级

Abstract:

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.

Key words: linear differential equations, transcendental integral function, hyper-order, the growth order of solution

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