常微分方程第二章课件

2first-orderdifferentialequation,2.3linearequations,2.2separablevariables,2.1Solutioncurveswithoutthesolution,上页,下页,铃,结束,返回,首页,2.4exactequation,2.5solutionsbysubstitutions,Ifwecanneitherfindnorinventamethodforsolvingit,2.1Solutioncurveswithoutthesolution,analytically,itisoftenpossibletogleanuseful,Informationaboutthenatureofsolutionsdirectlyfrom,Thedifferentialequationitself.,Integralcurves,一条曲线，称为微分方程的积分曲线。,方向场。,在方向场中，方向相同的点的几何轨迹为等斜线。,充分接近的值，就可得到足够密集的等斜线族。,2.2separablevariables,Thedifferentialequation,Solutionbyintegration,Whenfisindependentofthevariabley-thatis,canbesolvedbyintegration.Ifg(x)isacontinuous,function,thenintegratingbothsidesof(1)givesthe,solution,whereG(x)isanantiderivative(indefiniteintegral),ofg(x).,Definition2.1separableequation,Afirst-orderdifferentialequationoftheform,issaidtobeseparableortohaveseparable,variables.,Forexample,theequation,areseparableandnonseparable,respectively.,whereH(y)andG(x)areantiderivativesof,Observethatbydividingbythefunctionh(y)we,canwriteaseparableequationas,andtherefore,andg(x),respectively.,Notethereisnoneedtousetwoconstantsinthe,Methodofsolution,Aone-parameterfamilyofsolutions,usuallygiven,implicitly,isobtainedbyintegratingbothsidesof,integrationofaseparableequation.,Integratedbybothsides,Example1solve,solutionrewritingtheequation,get,cisanarbitraryconstant,Example2solvetheinitial-valueproblem,Solutionpleaseseparatethevariables,fromwhichitfollowsthat,thegeneralsolution,Integratedbytheboth,get,arbitraryconstant,that,Otherwise,theequationalsohavethesolution,Inordertodeterminetheparticularsolution,put,Therefore,theparticularsolutionis,intotheordinarysolutioninordertodefine,integratedbythebothsides,Example3findthegeneralsolutionof,P(x)iscontinuousfunctionofx.,There,Solutionseparatethevariables,get,isanarbitraryconstant.,Otherwise,obviouslyy=0alsoisthesolution,Fromthedefineoflog,weget,of(2).If(3)mayletc=0,they=0isinthe(3),Let,have,therefore,thegeneralsolutionof(2)is(3),therecisanarbitraryconstant.,Weonlyintroducetwotypes,2reductiontoseparationofvariables,Iscalledhomogeneousequation,g(u)iscontinuous,1)theform,functionofu.,Thisisaseparableequation.,Theaboveequationisaseparableequation.,Ifexistu=u0,andg(u0)-u0=0.Wecanhavetheresult,u=u0,isasolutionoftheaboveequation,soy=u0,isasolutionofthepreviousequation.,Solution,Example1solvetheequation,puttheaboveintotheequation,get,Let,Lettheaboveseparatevariableandintegrate,Have,Otherwise,theequationhavesolutiontgu=0,thatistosaysinu=0,thereforethesolutionis,Exercisesolvetheequation,alsocantransformtotheseparatevariable,2)theform,separatelyareconstants.,equation,Wediscussitbythreecases.,(1)thecaseof,Weneedusethesubstitution,theequation(1)cantransformintoseparate,variableequation.,(2)theform,Supposetheratioisk,thatistosay,Thentheequationcanbe,Let,Thisisaseparablevariableequation.,theequationbecometo,(3)theform,if,Andtheequation(1)becomeinto,Then,Put,Solutionwesolvethegroupofequation,Example2solvetheequation,intotheequation(2),get,if,solutionof(2)is,Alsoisthesolution.therefore,thegeneral,Weeasilyaffirm,There,cisanarbitraryconstant.,Prooftheequation,andsolvethefollowingequations,separableequationbytranslation,canbetranslated,Linearfirst-orderDE,2.3linearequations,thefirstdegreeinthedependentvariableandallits,Adifferentialequationissaidtobelinearwhenitisof,derivatives.Whenn=1in(6)ofSection1.1,weobtain,alinearfirst-orderdifferentialequation.,Afirst-orderdifferentialequationoftheform,Linearequation,Wheng(x)=0,thelinearequationissaidtobe,issaidtobealinearequation.,homogeneous(齐次的);otherwise,itis,Non-homogeneous（非齐次的）.,a1(x),weobtainamoreusefulform,thestandardform,Bydividingbothsidesof(1)bytheleadcoefficient,Weseekasolutionof(2)onanintervalIforwhich,ofalinearequation:,bothfunctionsPandfarecontinuous.,Theprocedure,Weknowthesolutionof,Wesuppose:in(3)lettheconstantcbecomefunction,Is,Weusethesolutionof(3)toobtainthesolutionof(2).,c(x),inordertosatisfytheequation(2),andobtainc(x),differentiate,obtain,therefore,let,Let(4)and(5)into(2),get,Thatistosay,Integrate,Thereisanarbitraryconstant.Let(6)into(4),obtain,Thisisthegeneralsolutionof(2).,isvariationofparameter(常数变易法),Themethodoflettingconstantbecomeintofunction,Example1,solvethegeneralsolutionoftheequation,here,nisaconstant.,Solutiontheequationbecometo,First,solvethehomogeneouslinearequationof,from,wecangetthegeneralsolutionofhomogeneous,linearequation,Second,weusethemethodvariationofparameter,tosolvethenon-homogeneouslinearequation.C,changeintoundefinedfunctionc(x)ofx,thatis,Differentiate,get,Thatistosay,Integrated,get,Thesolutionofprimaryequationis,Example2solvetheequation,solutiontheprimaryequationisntlinearequation,ofy,butwecanchangeit,Xislookedasundefinedfunction,yislookedas,independentvariables.,First,solve,getthesolution,usethemethodofvariationparameter,clookedas,c(y),differentiateit,get,Thatistosay,Otherwise,thegeneralsolutionofprimaryequationis,Here,isanarbitraryconstant.,2.4exactequation（恰当方程）,differentialoffunctionoftwovariables,Ifz=f(x,y)isafunctionoftwovariableswithcontinuous,firstpartialderivativesinaregionRofthexy-plane,then,itsdifferentialis,Nowiff(x,y)=c,itfollowsfrom(1)that,Definition2.3exactequation（恰当方程）,AdifferentialexpressionM(x,y)dx+N(x,y)dyisan,exactdifferential（恰当微分）inaregionRofthexy-,planeifitcorrespondstothedifferentialofsome,functionf(x,y).Afirst-orderdifferentialequationofthe,form,issaidtobeanexactequationiftheexpressionon,theleft-handsideisanexactdifferential.,Forexample,isanexactequation,because,Notice,if,then,Theorem2.1criterionforanexactdifferential,（恰当微分的判定）,LetM(x,y)andN(x,y)becontinuousandhave,continuousfirstpartialderivativesinrectangular,regionRdefinedby,Thenanecessaryandsufficientconditionthat,beanexactdifferentialis,Methodofsolution,If,thenexistafunctionfforwhich,Theimplicitsolutionoftheequationisf(x,y)=c,Notice:theexpression,isIndependentofx.,Example1solvethegeneralsolutionof,Solution,equations,Therefore,theequationisexactequation.,Now,wefindfinordertosatisfythefollowingtwo,Integrate(1)aboutx,get,Takingthepartialderivativeofthelastexpression,withrespecttoyandsatisfy(2),Therefore,thegeneralsolutionis,cisanarbitraryconstant.,“分项组合”：先把那些已构成全微分的项分出，再把,剩下的项凑成全微分，又叫“凑微分”。应熟记一些简,简单的二元函数的全微分，,Example2solvetheequationofexample1,Solution,Thentheordinarysolutionis,There,Cisanarbitraryconstant.,Example3solvetheequation,Solution,because,theequationisanexactequation.,Integratingfactors（积分因子）,Foranon-exactdifferentialequation,beanintegratingfactorof(4)is,itissometimespossibletofindanintegratingfactor,Isanexactdifferential.,Thenanecessaryandsufficientconditionthat,aftermultiplying,theleft-handsideof,Thatis,If,theequation(5)become,onlyonxis,isafunctionofxalone.,Thenanecessaryandsufficientconditionofdepends,Wecanobtaintheintegratingfactorof(4)is,Example1,thenonlinearfirst-orderdifferentialequation,Theintegratingfactoristhen,isnotexact.Withtheidentifications,AfterwemultiplythegivenDEby,theresultingequationis,Thatafamilyofsolutionsis,usingthemethodofintegratingfactortosolvethe,Example2,linearequation,and,Solutiontheequation(6)becometo,Have,Wemultiplytheequation(7)by,theresultingequationis,2.5solutionsbysubstitutions,Ifafunctionfpossessestheproperty,isahomogeneousfunctionofdegree3.,forsomerealnumbera,thenfissaidtobea,Forexample,Therefore,thefunction,homogeneousfunctionofdegreea,Afirst-orderDEindiffer