论文-二阶常微分方程解存在问题.docVIP

2012届毕业生 毕业论文 题 目: 二阶常微分方程解的存在问题分析 院系名称： 专业班级： 学生姓名： 学 号： 指导教师： 教师职称： 2012年5月25日  摘 要 在科学研究、工程技术中，常常需要将某些实际问题转化为二阶常微分方程问题。因此，研究不同类型的二阶常微分方程的求解方法及探讨其解的存在唯一性问题，是十分重要的。本文首先介绍了二阶常系数齐次线性微分方程的一般解法——特征方程法，及二阶常系数非齐次线性微分方程的待定系数法，然后又介绍了一些可降阶的微分方程类型。接着讨论了二阶变系数微分方程的幂级数解法，并论述了如何利用变量代换法将某些变系数方程化为常系数方程。另外，本文还介绍了求解初值问题的另一种方法——拉普拉斯变换法。最后，给出了二阶微分方程的存在唯一性定理的证明以及它的一些应用。 关键词：二阶线性微分方程，常系数，变系数，通解，特解，存在唯一性 Title: Analysis of the solution existence problem for Second order ordinary differential equation Abstract In science, engineering technology, we often need to conversion some practical problems into second-order ordinary differential equations. Therefore, to study the methods of different types of second order ordinary differential equation and to investigate the existence and uniqueness of the solution is very important. The paper first introduces the general solution of the second order constant coefficient homogeneous linear differential equations - the characteristic equation method, and the method for solving second order constant coefficient non-homogeneous linear differential equation undetermined coefficients, then describes some types of differential equation which can be reduced -order . Followed it, this article also discusses the utilization of power series solution to solve the second-order variable coefficient differential , and discusses how to use variable substitution to conversion certain equations with variable coefficients into constant coefficient. In addition, the article also describes another method for solving initial value problems - Laplace transform method. Finally, there gives the proof of the existence and uniqueness theorem of the second-order differential equations as well as some of its applications. Keywords:Second order linear differential ,Constant Coefficients ,Variable Coefficients ,General solution ,Particular solution ,Existence and Uniqueness .  目 录 §1 引言 5 §2 常系数线性微分方程的解法 5