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1、August7,201221:03c02Sheetnumber1Pagenumber31cyanblackCHAPTER2FirstOrderDifferentialEquationsThischapterdealswithdifferentialequationsoffirstorderdy=f(t,y),(1)dtwherefisagivenfunctionoftwovariables.Anydifferentiablefunctiony=φ(t)thatsatisfiesthisequationforalltinsomeintervaliscalledasolution,ando
2、urobjectistodeterminewhethersuchfunctionsexistand,ifso,todevelopmethodsforfindingthem.Unfortunately,foranarbitraryfunctionf,thereisnogeneralmethodforsolv-ingtheequationintermsofelementaryfunctions.Instead,wewilldescribeseveralmethods,eachofwhichisapplicabletoacertainsubclassoffirstorderequations
3、.Themostimportantofthesearelinearequations(Section2.1),separableequa-tions(Section2.2),andexactequations(Section2.6).Othersectionsofthischapterdescribesomeoftheimportantapplicationsoffirstorderdifferentialequations,intro-ducetheideaofapproximatingasolutionbynumericalcomputation,anddiscusssometh
4、eoreticalquestionsrelatedtotheexistenceanduniquenessofsolutions.Thefinalsectionincludesanexampleofchaoticsolutionsinthecontextoffirstorderdifferenceequations,whichhavesomeimportantpointsofsimilaritywithdifferentialequationsandaresimplertoinvestigate.2.1LinearEquations;MethodofIntegratingFactorsI
5、fthefunctionfinEq.(1)dependslinearlyonthedependentvariabley,thenEq.(1)iscalledafirstorderlinearequation.InSections1.1and1.2wediscussedarestrictedtypeoffirstorderlinearequationinwhichthecoefficientsareconstants.31August7,201221:03c02Sheetnumber2Pagenumber32cyanblack32Chapter2.FirstOrderDifferentia
6、lEquationsAtypicalexampleisdy=−ay+b,(2)dtwhereaandbaregivenconstants.Recallthatanequationofthisformdescribesthemotionofanobjectfallingintheatmosphere.Nowwewanttoconsiderthemostgeneralfirstorderlinearequation,whichisobtainedbyreplacingthecoefficientsaandbinEq.(2)byarbitraryfunctionsoft.Wewillusua
7、llywritethegeneralfirstorderlinearequationinthestandardformdy+p(t)y=g(t),(3)dtwherepandgaregivenfunctionsoftheindependentvariablet.SometimesitismoreconvenienttowritetheequationintheformdyP(t)+Q(t)y=G(t),(4)dtwhereP,Q,andGaregiven.Ofcours
