几类变系数线性常微分方程的求解

Solutions for Several Classes of Linear Ordinary Differential Equations with Variable Coefficients

在科学研究、工程技术中 ,人们常会遇到二阶或高阶变系数线性微分方程 ,一般形式的这类方程 ,无法用初等积分法求解 ,也没有通用的一般性方法。但这类方程中的一些特殊类型仍可求解。为了满足理论研究和工程实践的需要 ,一直以来 ,人们用不同的方法在不断的探讨这一问题 ,极大地扩展了变系数线性微分方程的可积类型。借助双变换 -未知函数的线性变换和自变量的变换 ,将几类变系数线性微分方程化为常系数的线性微分方程 ,从而求得它们的通解 。

There are second order or high order differential equations with variable coefficients in scientific research and engineering technology. The generic equations can't be solved by elementary integration method, but the special ones can still be solved. In order to meet the requirements of theoretical studies and engineering practice, people have been making efforts to study it. By means of double transformation (linear transformation of unknown function and self-variable transformation), several classes of linear differential equations with variable coefficients are turned into linear differential equations with constant coefficients. Thus, general solutions of equations mentioned above can be obtained. Meanwhile, the famous Euler equations and some predecessor's results on this issue are extended.