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1、August7,201221:03c01Sheetnumber1Pagenumber1cyanblackCHAPTER1IntroductionInthischapterwegiveperspectivetoyourstudyofdifferentialequationsinseveraldifferentways.First,weusetwoproblemstoillustratesomeofthebasicideasthatwewillreturnto,andelaborateupon,frequentlythroughoutt
2、heremainderofthebook.Later,toprovideorganizationalstructureforthebook,weindicateseveralwaysofclassifyingdifferentialequations.Finally,weoutlinesomeofthemajortrendsinthehistoricaldevelopmentofthesubjectandmentionafewoftheoutstandingmathematicianswhohavecontributedtoit.T
3、hestudyofdifferentialequationshasattractedtheattentionofmanyoftheworld’sgreatestmathematiciansduringthepastthreecenturies.Nevertheless,itremainsadynamicfieldofinquirytoday,withmanyinterestingopenquestions.1.1SomeBasicMathematicalModels;DirectionFieldsBeforeembarkingonas
4、eriousstudyofdifferentialequations(forexample,byread-ingthisbookormajorportionsofit),youshouldhavesomeideaofthepossiblebenefitstobegainedbydoingso.Forsomestudentstheintrinsicinterestofthesubjectitselfisenoughmotivation,butformostitisthelikelihoodofimportantapplicationst
5、ootherfieldsthatmakestheundertakingworthwhile.Manyoftheprinciples,orlaws,underlyingthebehaviorofthenaturalworldarestatementsorrelationsinvolvingratesatwhichthingshappen.Whenexpressedinmathematicalterms,therelationsareequationsandtheratesarederivatives.Equationscontainin
6、gderivativesaredifferentialequations.Therefore,tounderstandandtoinvestigateproblemsinvolvingthemotionoffluids,theflowofcurrentinelec-triccircuits,thedissipationofheatinsolidobjects,thepropagationanddetectionof1August7,201221:03c01Sheetnumber2Pagenumber2cyanblack2Chapter1
7、.Introductionseismicwaves,ortheincreaseordecreaseofpopulations,amongmanyothers,itisnecessarytoknowsomethingaboutdifferentialequations.Adifferentialequationthatdescribessomephysicalprocessisoftencalledamath-ematicalmodeloftheprocess,andmanysuchmodelsarediscussedthrougho
8、utthisbook.Inthissectionwebeginwithtwomodelsleadingtoequationsthatareeasytosolve.Itisnoteworthythateventhesimplestdif
