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1、IntroductiontoDifferentialEquations1ThepurposeofDifferentialEquationswithMathematica,ThirdEdition,istwofold.First,weintroduceanddiscussthetopicscoveredintypicalundergraduateandbeginninggraduatecoursesinordinaryandpartialdifferentialequationsinclud-ingtopicssuc
2、hasLaplacetransforms,Fourierseries,eigenvalueproblems,andboundary-valueproblems.Second,weillustratehowMathematicaisusedtoenhancethestudyofdifferentialequationsnotonlybyeliminatingthecompu-tationaldif?culties,butalsobyovercomingthevisuallimitationsassociatedwit
3、htheexplicitsolutionstodifferentialequations,whichareoftenquitecomplicated.Ineachchapter,we?rstbrie?ypresentthematerialinamannersimilartomostdifferentialequationstextsandthenillustratehowMathematicacanbeusedtosolvesometypicalproblems.Forexample,inChapter2,wein
4、troducethetopicof?rst-orderequations.First,weshowhowtosolvecertaintypesofproblemsbyhandandthenshowhowMathematicacanbeusedtoassistinthesamesolu-tionprocedures.Finally,weillustratehowMathematicacommandslikeDSolveandNDSolvecanbeusedtosolvesomefrequentlyencountere
5、dequationsexactlyand/ornumerically.InChapter3wediscusssomeapplicationsof?rst-orderequa-tions.SinceweareexperiencedandunderstandthemethodsofsolutioncoveredinChapter2,wemakeuseofDSolveandsimilarcommandstoobtainsolutions.Indoingso,weareabletoemphasizetheapplicati
6、onsthemselvesasopposedtobecomingboggeddownincalculations.TheadvantagesofusingMathematicainthestudyofdifferentialequationsarenumerous,butperhapsthemostusefulisthatofbeingabletoproducethegraphicsassociatedwithsolutionsofdifferentialequations.Thisisparticularlybe
7、ne?cialinthediscussionofapplicationsbecausemanyphysicalsituationsare12Chapter1IntroductiontoDifferentialEquationsmodeledwithdifferentialequations.Forexample,wewillseethatthemotionofapendulumcanbemodeledbyadifferentialequation.Whenwesolvetheproblemofthemotionof
8、apendulum,weusetechnologytoactuallywatchthependulummove.Thesameistrueforthemotionofamassattachedtotheendofaspringaswellasmanyotherproblems.Inhavingthisability,thestudyofdifferentia
