利用微分算子法研究二阶齐次线性微分方程与Riccati方程通解之联系

Study on the Relation between the General Solutions of the Second Order Homogeneous Linear Differential Equation and the Ones of Riccati Equation by Differential Operator Method

文章应用微分算子法处理二阶变系数线性微分方程,揭示了二阶齐次变系数线性微分方程与Riccati方程的通解理论之间的联系,发现这两类方程之间的通解可以互相转化,同时给出转化的途径.最后,作为理论的应用分析了一些具体例子.

The second-order linear differential equations with variable coefficients are processed by differential operator method, and the relation between the general solutions of the second-order homogeneous linear differential equations with variable coefficients and the ones of Riccati equations is revealed. Then, the conclusion that the general solutions between the two equations can be transformed into each other is deduced and the way of transformation is given. Finally, some examples are given as theoretical applications.