非齐次微分方程的求解方法.doc

毕业设计(论文) 题目名称：非齐次微分方程的求解方法 院系名称：理学院 班 级：数学082 学 号：200800134205 学生姓名：李清雅 指导教师：钱德亮 2012年5月  非齐次微分方程的求解方法 Methods of solving Inhomogeneous differential Equations 院系名称：理学院 班 级：数学082 学 号：200800134205 学生姓名：李清雅 指导教师：钱德亮 2012年 6 月 摘 要 虽然非齐次线性微分方程的求解是比较复杂的问题,但是对特定的方程还是有一定的解决方法.关于齐次线性微分方程的研究比较容易,但对于非齐次方程则要复杂的多.由于非齐次微分方程的解与齐次方程的解联系紧密,所以本文简单介绍了关于齐次微分方程与非齐次微分方程解的基础知识的基础上，通过分析,归纳总结出了一阶、二阶和n阶非齐次微分方程的解法.如常数变易法、比较系数法、简化待定系数法、算子法和叠加法等方法.通过对这些解法的研究,我们能清晰的理解各种解法适用的具体情况.由这些非齐次微分方程的解法,Abstract Although the solutions of the linear inhomogeneous differential equations are more complicated, there are still some special methods for specific equations. It is well known, it is relatively easy for solving to the linear homogeneous differential equations, but for linear inhomogeneous equations, it is far more complicated.. Because the solution of homogeneous differential equation and the inhomogeneous equation closely linked, so this paper simply introduces the homogeneous differential equation about homogeneous differential equations of the basic knowledge of basis, through analysis, this paper concludes that the first and second order and n order inhomogeneous differential equation solution. As usual constant variation, comparative coefficient method, simplify the method of undetermined coefficients, operator method and superposition method, etc. Through the method of the research, we can clear understanding of the various solutions of the specific conditions of the applicable. By these inhomogeneous differential equation of the basic method, so that we can better learning and understanding differential equation. Key Words: Differential equations; Inhomogeneous; Methods for solving目录 摘 要 I ABSTRACT II 1引言 1 2基本知识 2 2.1一阶非齐次线性微分方程的基本概念和定理 2 2.2二阶常系数非齐次微分方程的基本概念和定理 2 2.3 n阶常系数非齐次线性微分方程基本概念和定理 3 3非齐次微分方程的求解方法 5 3.1 一阶非齐次线性微分方程的求解方法 5 3.2二阶常系数非齐次微分方程的求解方法 7 3.3 n阶常系数非齐次线性微分方程求解方法 9 3.3.1 常数变易法 9 3.3.2 比较系数法 10 3.3.3 简化待定系数法 11 3.