1、高阶线性微分方程的积分因子解法 1 高阶线性微分方程的积分因子解 摘 要 ： 本文首先讲述了微分方程的基础知识，介绍了一些低阶常微分方程的已知的常用解法，并由此引入了积分因子解法的概念，及在一阶线性微分方程求解中的简便性 ,接着讲述了二阶线性微分方程的几种常用解法，在考虑这几种解法局限性的同时，引入了积分因子解法对二阶变系数线性微分方程进行求解，并取得了很好的效果。在此基础上，我们把积分因子解法扩展高阶线性微分方程中去。我们希望找到合适 的 2 3 112 ( ) ( )() , , , ,n n nx x xxe e e e ，使其满足 311 2 2 1) ( )( ) ( ( ( ( (
2、 ) ) ) ) ) ( )n n nx x xxxe e e e y e f x ， 则我们便可以得到 n 阶线性微分方程的通解： 1 1 2 1() () 12( ( ( ( ( ) ) ) ) )n n nxx xx ny e e e e f x d x c d x c d x c 。 由此便希望将此种方法也应用到欧拉方程中，找到欧拉方程形如 2 3 112 111 , , , ,n n nx x x x 的积分因子，使其满足 311 2 2 111( ( ( ( ( ) ) ) ) ) ( )n n nx x x x y x f x 从而可得通解为 1 1 2 1( 1 ) ( 1 )
3、 1 12( ( ( ( ( ) ) ) ) )n n n ny x x x x f x d x c d x c d x c 。 关键词 ： 高阶线性微分方程 ,n 阶常系数线性微分方程， 欧拉方程，积分因子 数学系 信息与计算科学 04-2 班 熊南南 20044544 2 Abstract: In this paper, at first we recount the foundational knowledge of the differential equation We introduce some methods which are known by most of us to s
4、olve some low-order differential equations, and then led to the notion of integral factor method It is convenient to use the integral factor to solve the first order linear differential equations Then we related some common methods to explain the second-order linear differential equations In the mea
5、ntime, considering the limit of these several kinds of solution methods, we led to go into integral factor method to the second order variable coefficient linear differential equation, and obtain good result. On this foundation, we expand this method to the nth-order ordinary coefficient of linear d
6、ifferential equations Suppose 2 3 112 ( ) ( )() , , , ,n n nx x xxe e e e are integral factors of a n-th order differential equation, make them satisfy： 311 2 2 1) ( )( ) ( ( ( ( ( ) ) ) ) ) ( )n n nx x xxxe e e e y e f x we then get the general solution of the nth-order linear differential equation
7、 , which is 1 1 2 1() () 12( ( ( ( ( ) ) ) ) )n n nxx xx ny e e e e f x d x c d x c d x c Following that we hope expand this method to Eulers equation, suppose 2 3 112 111 , , , ,n n nx x x x are integral factors of n-th Eulers equation, if they satisfy ： 311 2 2 111( ( ( ( ( ) ) ) ) ) ( )n n nx x x
8、 x y x f x then we get the general solution to the nth-order Eulers equation , which is 1 1 2 1( 1 ) ( 1 ) 1 12( ( ( ( ( ) ) ) ) )n n n ny x x x x f x d x c d x c d x c Keywords: Linear Equations of Higher Order， the nth-order Ordinary Coefficient of Linear Differential Equations, the nth-order Eulers Equation ,Integral Factor
