常系数线性常微分方程组的显式解
The Explicit Solution of Homogeneous Linear Ordinary Differential Equations with Constant Coefficients
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摘要: 本文利用张量分析给出了常系数线性常微分方程组和n阶常系数线性常微分方程初值问题一般解的显式表示,包括特征根有重根时的情况.实际上本文给出了计算矩阵exp[At]的元素的一般公式.这种方法不仅在公式表示上简洁方便,而且更适用于计算机的程序设计,大大加快了运算速度.Abstract: In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneous linear differential equation with constant coefficients.In fact,we give the general formula for calculating the elements of the matrix eAt.We also give the results when the characteristic equation has the repeated roots.The present method is simpler and better than the other methods.
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Key words:
- tensor differential equation /
- characteristic tensor /
- explicit solution
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[1] 王柔怀、伍卓珊,《常微分方程讲义》,人民教育出版社(1963).146. [2] 王高维、周之铭等,《常微分方程》(第2版),高等教育出版社(1983),223. -
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