高阶变系数线性齐次微分方程内xveλx型解的探求

高阶线性微分方程解的结构理论已很完善，但对一般变系数线性齐次微分方程至今尚未见到探求特解的有效方法．为了更多地得到在理论上和应用上占有重要地位的高阶线性微分方程的通解，对一般变系数高阶线性齐次微分方程引入特征多项式和特征方程的概念，运用高阶导数法则及高次代数方程的重根理论，得到了高阶变系数线性齐次微分方程内有x^veλx型解的一个新的、实用的充分判据，为探求一般变系数线性齐次微分方程内x^veλx型解提供了一个有效的方法，推广了经典的高阶常系数线性齐次微分方程的解法及一些近代的可解结果．

A Research on the xveλx Type Solution of Linear Homogeneous Differential Equation of Higher Order with Variable Coefficients

The structural theory of the solution of linear differential equation of higher order is rather perfect, but there is not any efficient method to obtain special solution by now. In order to obtain more general solution of linear differential equation of higher order which is important in theory and practice, the concepts of charactristic multinomial and characteristic equation are introduced for this kind of equation. By higher order derivation method and repeated roots theory of higher-degree algebraic equation, a practical sufficient criterion for the xvexl type solution was given. This provided an efficient method to searching the xvexl type solutions and extended the solution approaches of solving the classical linear differential equation of high order with constant coefficients and some modern solvable results.