一阶常微分方程初等解法.doc

目录摘要…………………………………………………………………………………......1关键词………………………………………...…………………………………………...1Abstract…………………………………………………………...………………………1Keywords………………………………………………………………………..………..10前言…………………..……………………………………………………………...11预备知识………………………………………………………………………..……...11.1变量分离方程……………………………………………………………………….21.2恰当微分方程……………………………………………………………………….21.3积分因子………………………………………………………………….…………22基本方法…………………………………………………………………………..…...22.1一般变量分离……………………………………………………………………….32.2齐次微分方程……………………………………………………………………….32.2.1齐次微分方程类型一………………………………………………………….32.2.2齐次微分方程类型二……………………..……...……………………………42.3常数变易法…………………………..………………...……………………………52.3.1常数变易法一…………………………….……………………………………52.3.2常数变易法二……………………….…………………………..……………..62.4积分因子求解法…………………………………..…………………………………72.5恰当微分方程求解法…………………………………………..……………………83基本方法的应用……………………………………………..………………………83.1一般变量分离方程应用…………………………………………….………………83.1.1应用举例……………………………………………………….………………93.1.2应用举例………………………………………...……..………………………93.2齐次微分方程应用……………………………………...…………………………103.2.1类型一应用举例…………………………………...…………………………103.2.2类型一应用举例………………………………………………………………113.2.3类型二应用举例………………………………………………………………113.2.4类型二应用举例………………………………………...……………………123.3常数变易法应用……………………………………………………………………133.3.1常数变易法应用举例……………………………………..…….……………133.3.2伯努利微分方程应用举例………………………………………...…………143.4利用积分因子求解……………………..……………………...……..……………143.5利用恰当微分方程求解……………………………………………....……..……15参考文献……………………………...………………..…………………......…………16一阶常微分方程初等解法摘要:本文对一阶微分方程的初等解法进行归纳与总结,同时简要分析了变量分离,积分因子,恰当微分方程等各类初等解法.并且结合例题演示了如何把常微分方程的求解问题化为积分问题,进行求解.关键词:一阶常微分方程;变量分离;恰当微分方程;积分因子TheFundamentalmethodsofthefirst-orderordinarydifferentialequationAbstract:Inthisthesis,wesummarizethefundamentalmethodsofthefirst-orderordinarydifferentialequation.Atthesametime,weanalysisthevarioustypesoffundamentalmethodssuchastheseparationofvariables,integratingfactorandtheexactdifferentialequation.Combinedwithexamples,weshowhowtheordinarydifferentialequationssolveproblemsbytransformingthemintotheproblemsofintegration.KeyWords:first-orderordinarydifferentialequation;separationofvariables;exactdifferentialequat