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山西师范大学本科毕业论文（设计）常微分方程的初等解法与求解技巧姓名张娟院系数学与计算机科学学院专业信息与计算科学班级12510201学号1251020126指导教师王晓锋答辩日期成绩常微分方程的初等解法与求解技巧内容摘要常微分方程在数学中发挥着举足轻重的作用，同时它的应用在日常生活里随处可见，因此掌握常微分方程的初等解法与求解技巧是非常必要的.本论文主要论述了其发展、初等解法与求解技巧，前者主要有变量分离、积分因子、一阶隐式微分方程的参数表示，通过举例从中总结出其求解技巧，目的是掌握其求解技巧.【关键词】变量分离一阶隐式微分方程积分因子求解技巧ElementarySolutionandSolvingSkillsofOrdinaryDifferentialEquationAbstractOrdinarydifferentialequationstakeupsignificantpositioninmathematics,andatthesametime,theapplicationofitcanbeseeneverywhereinourdailylife,therefore,it’snecessarytograsptheelementarysolutionofordinarydifferentialequationsandsolvingskills.Thispapermainlyintroducedthedefinitionofordinarydifferentialequations,elementarysolutionmethodandsolvingskills,theformermainlyincludedtheseparationofvariables,integralfactor,aparameter-orderdifferentialequationsimplicitrepresentation,bywayofexamplestosumuptheirsolvingskills,thepurposeistomastertheskillstosolve.【KeyWords】theseparationofvariablesthefirstorderimplicitdifferentialequationintegratingfactorsolutiontechniques目录1.引论·····························································································································12.变量分离方程与变量变换·············································································12.1变量分离方程的解法······························································································12.2变量分离方程的举例······························································································22.3变量分离方程的几种类型······················································································23.线性微分方程和常数变易法········································································63.1线性微分方程与常数变易法·················································································63.2伯努利微分方程·····································································································84.恰当微分方程与积分因子·············································································94.1恰当微分方程·········································································································94.2积分因子···············································································································115.一阶隐式微分方程与参数表示································································135.1一阶隐式微分方程的主要类型···········································································136.常微分方程的若干求解技巧······································································186.1将一阶微分方程dxdy变为dydx的形式···································································186.2分项组合···············································································································196.3积分因子的选择···································································································207.总结···························································································································21参考文献························································································错误!未定义书签。致谢································································································································221常微分方程的初等解法与求解技巧学生姓名：张娟指导教师：王晓锋1.引论常微分方程的实质就是一个关系式，这个关系式是由自变量、未知函数和未知函数的导数组成的，且自变量的个数为一个.其发展历史经历了一个很漫长的过程，在这个发展过程中涌现出很多科学家例如欧拉、拉格朗日、柯西等，他们对常微分方程的发展做出了很大的贡献.常微分方程的发展历史可分为三个阶段，分别是“求通解”阶段、“求定解”阶段、“求所有解”的新阶段.常微分方程在数学中占有很重要的地位，有很多伟人例如赛蒙斯都曾评价过常微分方程在数学中的地位，指出其在数学中的不可替代的作用.常微分方程非常重要，其初等解法有很多种，我们应该掌握其初等解法与技巧.2.变量分离方程与变量变换2.1变量分离方程的解法对于变量分离方程)()(yxfdxdy，若0)(y，则有：dxxfydy)()(，两边积分，得到:cdxxfydy)()(，c为任意实数.如果0)(y得0yy，验证一下0yy是否包括在cdxxfydy)()(中，若不包括，需补上特解0yy.22.2变量分离方程的举例（1）xydxdy2，求该方程的解．解：当0y时，xdxydy2，两边积分，得到：12cxdxydy，1c为任意实数．故2xcey，c为任意实数．显然y=0包括在2xcey中，故方程的通解为：2xcey，c为任意实数．2.3变量分离方程的几种类型2.3.1齐次微分方程对于齐次微分方程)(xygdxdy，解法：令xyu则有：uxy，（2-1）两边对x求导得：udxduxdxdy，（2-2）将（2-1），（2-2）代入齐次微分方程)(xygdxdy中可得：)(ugudxdux，即xuugdxdu)(，从而可以求得其解．举例：求解方程)0(2xyxydxdyx.解：原方程可化解为：xyxydxdy20x，3这个方程为齐次微分方程，令uxy，则有xuy，两边对x求导得：udxduxdxdy，将uxy和udxduxdxdy代入原方程中得：udxdux2，这个方程为可分离变量方程，当0u时解之可得：cxu)ln(,其中c为使等式有意义的任意常数.即当0u时，显然是udxdux2的解，且不包含在cxu)ln(中，将uxy代入0u或cxu)ln(中可得：,0,0)(ln,])[ln(2cxcxxy当2.3.2有理比式222111cybxacybxadxdy的三种类型①类型一2121bbaakcc21（常数）情形，则原方程变为：kdxdy，故方程的通解为：ckxy，其中c为任意常数.举例：求解下列方程的解12224yxyxdxdy.解：根据题意可得：212224yxyxdxdy，即2dxdy，故可得：cxy2，c为任意常数.因此原方程的通解为：4cxy2，c为任意常数.②类型二212121cckbbaa情形，令ybxau22，两边对x求导可得：212222cuckubadxdybadxdu，这个方程是变量分离方程.举例：做适当变换求解方程25yxyxdxdy.解：经判断为第二种类型，令yxu，两边对x求导可得：dxdydxdu1，故可得:27udxdu，解之可得：127221cxuu，1c为任意常数.将yxu代入并化简可得：cxyxyyx104222，c为任意常数.③类型三2121bbaa情形，如果方程222111cybxacybxadxdy中的1c，2c不全等于零，111cybxa，222cybxa都是x，y的一次多项式，则,0,0222111cybxacybxa（2-3）可以求得解为：,,yx令,,yYxX5则（2-3）化解为：,0,02211YbXaYbXa故222111cybxacybxadxdy化为:)(2211XYgYbXaYbXadXdY，故可以解出该方程的解，解出其解，再将,,yYxX带入其解中，从而得到所求方程的解.举例：解下列方程1212yxyxdxdy.解：显然2121bbaa，故为第三种类型，解方程组3131yYxX得：31x，31y.于是令,