摘要
讨论实际问题中一类二阶变系数线性齐次常微分方程的数学模型,利用幂级数待定系数法得到了一般情况下的幂级数解的形式.在特殊条件下,对相应系统做变换,并利用变量分离法得到具有初等函数形式的解析解,并分析了在此情况下解的渐近性.最后,利用Lyaponov方法进行渐近性分析,得到了在一定条件下的收敛性结果,这个渐近性收敛结果在实际应用中是存在的,与某种特殊条件下的解收敛性相一致,从而说明了该数学模型在应用上有一定实际意义.
The mathematical model of a class of Second-order linear homogeneous ordinary differential equations with variable coefficients is investigated.By using the undetermined coefficient method of power series,the general form of power series solution is obtained.Under special conditions,by the transformation of the functions,the analytic solutions with the form of elementary function is obtained via the method of separation of variables.In addition,the asymptotic behaviors of the solution in this case are analyzed.The asymptotic properties of this problem are carried out by using Lyaponov method.And the convergence of solution is given under some conditions.The asymptotic convergence under some special conditions is consistent with the practice problems,which shows that the mathematical model has certain practical significance in application.
作者
郭春晓
郭艳凤
徐剑琴
GUO Chun-xiao;GUO Yan-feng;XU Jian-qin(School of Science,China University of Mining and Technology Beijing,Beijing 100083,China;School of Science,Guangxi University of Science and Technology,Liuzhou 545006,China;School of Mathematics and Physics,China University of Geosciences,Wuhan 430074,China;Engineering Training Center,Guangxi University of Science and Technology,Liuzhou,Guangxi 545006,China)
出处
《数学的实践与认识》
北大核心
2020年第5期323-328,共6页
Mathematics in Practice and Theory
基金
广西科技大学优秀教学团队和中国矿业大学(北京)本科教育教学改革与研究项目资助
国家自然科学基金(11861013,11771444,61663003)
广西自然科学基金项目(2017GXNSFAA198221)。
关键词
变系数常微分方程
幂级数
Lyaponov方法
渐近性分析
ordinary differential equations with variable coefficients
power series
Lyaponov method
asymptotic analysis
