常微分方程课件-第02讲

Lecture2,1,4.SolutionofODE(常微分方程的解),Definition1.4C/PP17-19,E/P6,Forannth-orderODE,SupposethereareanintervalIandafunctiony=(x)definedonI,satisfyingthefollowingconditions,Thenthefunctiony=(x)issaidtobeasolutionof(1.38)onI.,2,3,3,Example1.2,wherecisanarbitraryconstant.,Example1.3,4,Remark:Classificationofsolutions(解的分类),(1)ExplicitSolutionandImplicitSolution(显式解与隐式解),Definition1.5C/P17,E/P33,5,Inthiscourse,wedontdistinguishtheimplicitsolutionortheexplicitsolution,andtheyarebothcalledthesolutionofODE!(在本课程中，我们不区分隐式解与显式解，且它们统称为常微分方程的解。SeeC/P18),Remark:Classificationofsolutions(解的分类),6,(2)GeneralSolutionandParticularSolution(通解与特解),(i)GeneralSolution,Definition1.6C/P18,E/P10,34,102,103,Remark:Classificationofsolutions(解的分类),7,Itisageneralsolution,Itisnotageneralsolution,Why?,(2)GeneralSolutionandParticularSolution(通解与特解),8,Question:,(2)GeneralSolutionandParticularSolution(通解与特解),9,Proposition(Cf.C/PP23-24):,(2)GeneralSolutionandParticularSolution(通解与特解),10,Example1.4,Questions:,(1)HowdowedefinetheimplicitgeneralsolutionofODE?,(2)WhetherdoesageneralsolutionofanODEcontainallthesolutionofit?(SeeE/P11),AlsoseeC/PP35-36+PP63-65,(2)GeneralSolutionandParticularSolution(通解与特解),Proof:,11,(ii)Particularsolution,Conditionsfordeterminingsolution(定解条件)meanstheconditionsthatsomesolutionofODEmustsatisfy.(C/P18),Initialconditionsofnth-orderODE(n阶常微分方程的初始条件)meansthefollowingnconditions:(C/P18),(2)GeneralSolutionandParticularSolution(通解与特解),12,Problemfordeterminingsolution(定解问题)meanstofindasolutionofODEwhichsatisfiestheconditionsdeterminedsolution.,Initial-valueproblem(初值问题)meansthattheconditionsdeterminedsolutionistheinitialconditions.,Thegeneralformofinitial-valueproblemofnth-ordercanbewritten,Note:,(1)Initial-valueproblemissometimescalledCauchyproblem.,(2)Boundary-valueproblemC/P370.,(2)GeneralSolutionandParticularSolution(通解与特解),AparticularsolutionofODEmeansthatasolutionsatisfyinggiveninitialconditions.C/P18,E/P10,Ingeneral,aparticularsolutionofODEcanbeobtainedfromthegeneralsolutionofit.,13,Example1.5,Trytofindtheparticularsolutionof,14,Solution:,ElementaryMethodsofSolutionfor1st-orderODE,(一阶微分方程的初等解法),Chapter2(第二章),15,ElementaryMethodsofSolutionforDE?,微分方程的初等解法?,C/P30,所谓微分方程的初等解法是指把微分方程的求解问题化为积分问题，其解的表达式由初等函数或超越函数表示。,16,Aconvention(一个约定),Forexample,or,Forexample,wherecisanarbitraryconstant.,17,2.1EquationsofSeparatedVariablesandChangeofVariables(变量分离方程与变量变换),I.EquationsofSeparatedVariables,The1st-orderODE,iscalledseparable(可分离)providedthatg(x,y)canbewrittenastheproductofafunctionofxandafunctionofy:,Definition2.1.1C/P30,E/P31,Thus,y=g(x,y)becomesthefollowingform,18,Question:,TrytodeterminewhichoneofthefollowingODEsisseparable,whichoneisnot?,Answer,Yes,No,Yes,No,No,Yes,19,MethodofSolution(解法),equationseparatedvariables(已分离变量方程),Proposition(命题)C/P31,E/P31,i.e.,wherecisanarbitraryconstant.,(1),equationofseparablevariables(可分离变量方程),(TheproofisgiveninE/P32).,20,Twopossiblecases:,Noproblem,Inthiscase,whenfindingallsolutionsofit,wemustmakeitup.Sometimessuchasolutionisalsocalledanexceptionalsolution(例外解)oraconstantsolution(常数解).,(2),21,Why?,Remark:,22,Example2.1.1E/P33/Example2,Solution,Afterseparatingthevariablesandintegratingbothsides,weget,Thus,this(function)equationisthedesired(implicit)generalsolutionofthegivenODE,wherecisanarbitraryconstant.,23,Example2.1.2E/P35/Example4,Solution,Inaddition,Therefore,thedesiredsolutionsare,thegeneralsolution:,theexceptionalsolution:,24,Example2.1.3,Solution,Integratingbothsides,weget,Takingexponentsofbothsidesofthisequationgives,Inaddition,anditcanbeobtainedbyselectingc=0fromthegeneralsolution.,Therefore,SeeC/P33,E/P32,25,Warning!(警告!),FindthegeneralsolutionofanODEisdifferentfromthatFind(all)thesolutionsofanODEorSolveanODE!,求常微分方程的通解与求常微分方程的(所有)解或解常微分方程是不同的！,26,Remark:,Forexample:,27,II.SometypesofODEwhichcanbechangedintoequationofseparatedvariables(可化为变量分离方程的类型)C/P34,1.HomogeneousEquationsC/P34,E/P58,齐次方程,Definition2.1.2,(HomogeneousFunction)数学分析下P132/7,齐次函数,28,Forexample,29,Proposition：,or,30,Proof：,31,Definition2.1.3,(HomogeneousEquation),齐次方程,Forexample,32,MethodofSolutionofHomogeneousEquationC/P34,E/P58,Then,ThisisaseparableODE,33,Question:,34,Example2.1.4C/P34,Solution,Theseyield,Therefore,thedesiredgeneralsolutionis,35,Example2.1.4C/P34,36,Example2.1.5C/P35,Solution,Why,Example2.1.5C/P35,Solution,Let,then,Theseyield,Inaddition,butitcannotbeobtainedfromthegeneralsolution.,Thus,thedesiredsolutionare,ageneralsolution:,theexceptionalsolution:,and,38,Example2.1.6,Solution,Let,then,Thus,thedesiredgeneralsolutionis,cfC/P73/1(21),39,1.HomogeneousEquations,2.ODEoftheformcfC/P37,(*)isahomogeneousequation!,(*)isnotahomogeneousequation!,ItisOK!,II.SometypesofODEwhichcanbechangedintoequationofseparatedvariables(可化为变量分离方程的类型)(C/P34),40,Thisisahomogeneousequation,itsOK!,Thisisaseparableequation,itsOK!,41,Example2.1.7,Solution,Theseyield,Bytheseparationofvariables,weget,i.e.,Thusthedesiredgeneralsolutionis,cfC/P38,42,Example2.1.8,Solution,Bytheseparationofvariables,weget,Thusthedesiredgeneralsolutionis,i.e.,Question:whyhavewenotdiscussedif5u+7is0?,43,Briefsummaryfor2.1,I.EquationofSeparatedVariables,II.SometypesofODEwhichcanbechangedintoequationofseparatedvariables,Ideal(思路),Bytryingtofindanexactchangeofvariables,Changetheequationwhichisnotseparabletothatwhichisseparable,通过设法寻找合适的变量变换，把一个(表面上)不是变量可分离的微分方程化为变量可分离的微分方程。,Itsveryeasy!,Itsverydifficult,andthereisnotgeneralresult!,cfC/P38;C/P43/2.,2.1EquationofSeparatedVariablesandChangeofVariable,44,2.1EquationofSeparatedVariablesandChangeofVariable,I.EquationofSeparatedVariables,II.SometypesofODEwhichcanbechangedintoequationofseparatedvariables,III.Applications(应用),45,Example2.1.9,CharginganddischargingforcapacitorC/P39,电容器的