一阶常微分方程的初等解法的讨论-毕业论文网

论文总字数：9543字

目录

0、引言 4

1、变量分离方程 5

2、常数变异法 8

3、积分因子法 15

4、恰当微分方程求解 18

5、参数法求解一阶隐式微分方程 21

6、两类常微分方程的典型证明 23

结论 25

参考文献 25

致谢 27

一阶常微分方程的初等解法的讨论

胡梦园

,China

Abstract: The solution of ordinary differential equations has been continuously innovating with the development of mathematical research, and it is also a difficulty in the process of learning ordinary differential equations. In this paper, the variable separation equation, the constant variation method, the integral factor method and the parametric method are introduced. Among them, various types of equations have been summarized and summarized, as well as individual personal experience. This article not only briefly explains basic knowledge, but also extends the order reduction of Bernoulli's higher order differential equations, transforms high-order differential equations into well-known first-order differential equations through reduced order, and then uses the constant variation method. Transform variable-coefficient differential equations into constant-coefficient differential equations. The solution to ordinary differential equations is actually to translate differential problems into integral problems. The elementary solution of first-order ordinary differential equations is ever-changing in practical applications. It is necessary to be familiar with the basic concepts, and to apply the methods flexibly on specific issues, so that the problems can be simplified and solved.

Key words: Ordinary differential equation; Variable separation equation; Integral factor method; Variation of constants method; Bernoulli equation

0前言

常微分方程是一个历史悠久且发展迅速的学科，它是在牛顿、莱布尼兹微积分创立之后，数学家们发现用简单的积分无法解决的一类专项技术问题，进而，常微分方程由此诞生。

常微分方程的发展大致分为四个阶段：经典阶段、适定性阶段、解析理论阶段、定性理论阶段。经历了长久的发展，数学家们智慧的积淀，还有如今学者的整理和总结，就得出了我们当下所学的常微分方程。

一阶常微分方程是微分方程中最为基础的一类方程，但是其求解方法却是最为繁多且精辟。同时，高阶常微分方程的求解也是通过各种手段进行降阶，从而转化为一阶常微分方程来求解。

本文对一阶常微分方程的初等解法进行了简单的介绍，并加以延伸和应用。同时，对于周期解和解的有界性给出了两个典型的证明。

1变量分离方程

1.1.形如

（1.1）