求解常微分方程的若干技巧张轩瑜目录TOC\o"1-3"\h\z\uHYPERLINK\l"_Toc38317502"摘要PAGEREF_Toc38317502\h1HYPERLINK\l"_Toc38317503"关键词PAGEREF_Toc38317503\h1HYPERLINK\l"_Toc38317504"AbstractPAGEREF_Toc38317504\h2HYPERLINK\l"_Toc38317505"KeywordsPAGEREF_Toc38317505\h2HYPERLINK\l"_Toc38317506"一、引言PAGEREF_Toc38317506\h3HYPERLINK\l"_Toc38317507"二、常微分方程的几种类型及其定义PAGEREF_Toc38317507\h3HYPERLINK\l"_Toc38317508"三、一阶微分方程解法PAGEREF_Toc38317508\h4HYPERLINK\l"_Toc38317509"3.1变量分离法PAGEREF_Toc38317509\h4HYPERLINK\l"_Toc38317510"3.2常数变易法PAGEREF_Toc38317510\h8HYPERLINK\l"_Toc38317511"3.3积分因子法PAGEREF_Toc38317511\h10HYPERLINK\l"_Toc38317512"四、高阶微分方程的解法PAGEREF_Toc38317512\h14HYPERLINK\l"_Toc38317513"4.1特征根法PAGEREF_Toc38317513\h14HYPERLINK\l"_Toc38317514"4.2比较系数法PAGEREF_Toc38317514\h15HYPERLINK\l"_Toc38317515"4.3拉普拉斯变换法PAGEREF_Toc38317515\h17HYPERLINK\l"_Toc38317516"总结PAGEREF_Toc38317516\h19HYPERLINK\l"_Toc38317517"致谢PAGEREF_Toc38317517\h20HYPERLINK\l"_Toc38317518"参考文献PAGEREF_Toc38317518\h21PAGE\*MERGEFORMAT2求解常微分方程的若干技巧摘要:《微分方程》是自变量,未知函数及其导数的关系式，自变量只有一个的微分方程称为常微分方程。求解一阶微分方程的方法有许多技巧。本文主要总结了几种有初等解法的方程类型求解的一般方法。这些方法的基本原则就是将求解微分方程化为求积分问题,如:变量分离方程；线性微分方程；恰当微分方程等，同时本文也总结了求解高阶常系数微分方程的几种方法。关键词:变量分离方程；线性微分方程；恰当方程；伯努利方程；高阶常系数微分方程摘要关键词SometechniquesforsolvingordinarydifferentialequationsAbstract:Thedifferentialequationistherelationbetweenindependentvariables,unknownfunctionsandtheirderivatives,adifferentialequationwithonlyoneindependentvariableiscalledanordinarydifferentialequation.Therearemanytechniquesforsolvingfirstorderdifferentialequations.Thispapermainlysummarizesseveralgeneralmethodsforsolvingthetypesofequationswithelementarysolutions.Thebasicprincipleofthesemethodsistosolvethedifferentialequationintoanintegralproblem,forexample:variableseparationequation;lineardifferentialequation;exactdifferentialequationsandsoon,atthesametime,thispaper