常微分方程课后答案(第三版)王高雄

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1、习题3.31．。Proof若（1）成立则及，，使当时，初值问题的解满足对一切有,由解关于初值的对称性，（3，1）的两个解及都过点，由解的存在唯一性，当时故若（2）成立，取定，则，，使当时,对一切有因初值问题的解为，由解对初值的连续依赖性，对以上，，使当时对一切有而当时，因故这样证明了对一切有2．Proof：因及都在G内连续，从而在G内关于满足局部Lipschitz条件，因此解在它的存在范围内关于是连续的。设由初值和足够小）所确定的方程解分别为，即，于是因及、连续，因此这里具有性质：当时，；且当时，因此对有即是初值问题的解，在这里看成参数0显然，当时，上述

2、初值问题仍然有解。根据解对初值和参数的连续性定理，知是的连续函数，从而存在而是初值问题的解，不难求解它显然是的连续函数。3．解：这里满足解对初值的可微性定理条件故：满足的解为故4．解：这是在（1，0）某领域内满足解对初值可微性定理条件，由公式易见是原方程满足初始条件的解故AcknowledgementsMydeepestgratitudegoesfirstandforemosttoProfessoraaa,mysupervisor,forherconstantencouragementandguidance.Shehaswalkedmethroughal

3、lthestagesofthewritingofthisthesis.Withoutherconsistentandilluminatinginstruction,thisthesiscouldnothavereacheditspresentform.Second,IwouldliketoexpressmyheartfeltgratitudetoProfessoraaa,wholedmeintotheworldoftranslation.IamalsogreatlyindebtedtotheprofessorsandteachersattheDepart

4、mentofEnglish:Professordddd,Professorssss,whohaveinstructedandhelpedmealotinthepasttwoyears.Lastmythankswouldgotomybelovedfamilyfortheirlovingconsiderationsandgreatconfidenceinmeallthroughtheseyears.Ialsoowemysinceregratitudetomyfriendsandmyfellowclassmateswhogavemetheirhelpandti

5、meinlisteningtomeandhelpingmeworkoutmyproblemsduringthedifficultcourseofthethesis.MydeepestgratitudegoesfirstandforemosttoProfessoraaa,mysupervisor,forherconstantencouragementandguidance.Shehaswalkedmethroughallthestagesofthewritingofthisthesis.Withoutherconsistentandilluminating

6、instruction,thisthesiscouldnothavereacheditspresentform.Second,IwouldliketoexpressmyheartfeltgratitudetoProfessoraaa,wholedmeintotheworldoftranslation.IamalsogreatlyindebtedtotheprofessorsandteachersattheDepartmentofEnglish:Professordddd,Professorssss,whohaveinstructedandhelpedme

7、alotinthepasttwoyears.Lastmythankswouldgotomybelovedfamilyfortheirlovingconsiderationsandgreatconfidenceinmeallthroughtheseyears.Ialsoowemysinceregratitudetomyfriendsandmyfellowclassmateswhogavemetheirhelpandtimeinlisteningtomeandhelpingmeworkoutmyproblemsduringthedifficultcourse

8、ofthethesis.