一类二阶变系数常微分方程的解及其渐近性

Asymptotic behaviors of Solutions of Some Two Order Linear Ordinary Differential Equation with Variable Coefficients

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.