郑州航空工业管理学院
毕业论文（设计）
2015届数学与应用数学专业1111062班级
题 目 二阶常微分方程的降阶解法
姓 名贾静静学号111106213 指导教师程春蕊职称讲师
2015年4月5号
二阶常微分方程的降阶解法
摘 要
常微分方程是数学领域的一个非常重要的课题，并在实践中广泛于解决问题，分析模型。常微分方程在微分理论中占据首要位置,普遍应用在工程应用、科学研究以及物理学方面,不少应用范例都归结为二阶线性常微分方程的求解问题。而正常情况下，常系数微分方程依据线性常微分方程的日常理论是可以求解的.不过对于变系数二阶线性常微分方程的求解却有一定程度的困难,迄今为止还没有一个行之有效的普遍方法。
本文主要考虑了二阶常系数线性微分方程的降阶法。关于二阶常系数线性微分方程的求解问题,首先,我们给出二阶齐次常系数线性微分方程的特征方程，并求解出特征方程的两个特征根；其次，利用积分因子乘以微分方程和导数的运算，将二阶常系数线性微分方程化为一阶微分形式；最后，将一阶微分形式两边同时积分，求解一阶线性微分方程，可求得二阶常系数线性微分方程的一个特解或通解。关于二阶变系数齐次线性微分方程的求解问题，化为恰当方程通过降阶法求解二阶齐次变系数微分方程的通解。对于非齐次线性微分方程，只需再运用常数变易法求出它的一个特解，问题也就相应地解决了。
关键词
二阶常微分方程；降阶法；特征根；常数变易法；一阶微分形式
Order reduction method of second order ordinary differential equations
Jingjing Jia Chunrui Cheng 111106213
Abstract
Ordinary differential equation is a very important topic in the field of mathematics, it has been widely used in solving the problem and analyzing model in practice . Ordinary differential equations in the theory of differential occupied first place, it has been widely used in engineering application and scientific research as well as physics, many application examples are attributed to second order linear ordinary differential equation solving problem. And under normal circumstances,ordinary coefficient differential equation on the basis of the linear often daily theory of differential equations is can be solved. But for the solution for variable coefficient second order linear ordinary differential equations have a certain degree of difficulty, so far we haven't a well-established general method.
This paper mainly introduces the method of reduction of order two order linear differential equation with constant coefficients.On the problem of solving the linear differential equation with two order constant coefficients,first, we give homogeneous ordinary coefficient linear differential equation of the characteristic equation and solve the two characteristic roots of characteristic equation;secondly,we should use the integral factor times differential equation and derivative operation and turn two order constant
coefficient linear differential equation into the first order differential equation. Finally, We first order differential and integral form on both sides, solve the first order linear differential equations and find out a special solution or general solution of the second order linear constant coefficient differential equation. We solve the problem of second order homogeneous linear differential equation with variable coefficients, and should be turned into the appropriate equation, through the order reduction method to solve the second order homogeneous general solution of differential equation with variable coefficients.Solving non-homogeneous linear differential equation, we need to calculate it by applying the method of constant variation of a particular solution, problem is solved accordingly.
Keywords
second order ordinary differential equation ； Order reduction method; Characteristic root; Constant variation method; A first order differential form.
目 录
第一章 预备知识...........................................2 第二章 二阶常系数线性微分方程的降阶法.....................5 2.1提出问题........................................5 2.2二阶非齐次常系数线性微分方程的降阶法............6 2.3举例............................................6 2.4小结............................................8 第三章 二阶变系数线性常微分方程的降阶法...................9 3.1提出问题.......................................10 3.2二阶齐次变系数线性常微分方程的降阶法...........10 3.2.1解.............10
3.2.2求满足条件2的恰当方程的通解..............12 3.3小结...........................................14
求满足条件
1
的恰当方程的通
第四章 可降阶的二阶常微分方程............................15
d2y?f?x?型的微分方程..........................15 24.1 dx
d2y?dy??f?x,? 4.2 d2x?dx?型的微分方程.......................15 d2y?dy??f?y,?2 4.3 dx?dx?型的微分方程.......................16
第五章 可降阶的高阶常微分方程............................18
二阶常微分方程的降阶解法.
