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目 录

1 引言 ……………………………………………………………………………1 2 二阶常系数常微分方程的几种解法 …………………………………………1

2.1 特征方程法 ……………………………………………………………1 2.1.1 特征根是两个实根的情形 …………………………………………2 2.1.2 特征根有重根的情形 ………………………………………………2 2.2 常数变异法………………………………………………………………4 2.3 拉普拉斯变化法…………………………………………………………5 3 常微分方程的简单应用 ………………………………………………………6

3.1 特征方程法 ……………………………………………………………7 3.2 常数变异法………………………………………………………………9 3.3 拉普拉斯变化法…………………………………………………………10 4 总结及意义 …………………………………………………………………11 参考文献…………………………………………………………………………12

二阶常微分方程的解法及其应用

摘要：本文通过对特征方程法、常数变易法、拉普拉斯变换法这三种二阶常系数常微分方程解法进行介绍，特别是其中的特征方程法分为特征根是两个实根的情形和特征根有重根的情形这两种情况，分别使用特征值法、常数变异法以及拉普拉斯变换法来求动力学方程，现今对于二阶常微分方程解法的研究已经取得了不少成就，尤其在二阶常系数线性微分方程的求解问题方面卓有成效。应用常微分方程理论已经取得了很大的成就，但是，它的现有理论也还远远不能满足需要，还有待于进一步的发展，使这门学科的理论更加完善。

关键词：二阶常微分方程;特征分析法；常数变异法；拉普拉斯变换

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METHODS FOR TWO ORDER ORDINARY DIFFERENTIAL

EQUATION AND ITS APPLICATION

Abstract:This paper introduces the solution of the characteristic equation method, the method of variation of parameters, the Laplasse transform method the three kind of two order ordinary differential equations with constant coefficients, especially the characteristic equation method which is characteristic of the root is the two of two real roots and characteristics of root root, branch and don't use eigenvalue method, method of variation of constants and Laplasse transform method to obtain the dynamic equation, the current studies on solution of ordinary differential equations of order two has made many achievements, especially in the aspect of solving the problem of two order linear differential equation with constant coefficients very fruitful. Application of the theory of ordinary differential equations has made great achievements, however, the existing theory it is still far from meeting the need, needs further development, to make the discipline theory more perfect.

Keywords:second order ordinary differential equation; Characteristic analysis; constant variation method; Laplasse transform

1 引言

数学发展的历史告诉我们，300年来数学分析是数学的首要分支，而微分方程

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