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1、August7,201221:04c07Sheetnumber1Pagenumber359cyanblackCHAPTER7SystemsofFirstOrderLinearEquationsTherearemanyphysicalproblemsthatinvolveanumberofseparateelementslinkedtogetherinsomemanner.Forexample,electricalnetworkshavethischar-acter,asdomanyproblemsinmechanicsandinothe
2、rfields.Intheseandsimilarcases,thecorrespondingmathematicalproblemconsistsofasystemoftwoormoredifferentialequations,whichcanalwaysbewrittenasfirstorderequations.Inthischapterwefocusonsystemsoffirstorderlinearequations,andinpar-ticularequationshavingconstantcoefficients,utili
3、zingsomeoftheelementaryaspectsoflinearalgebratounifythepresentation.Inmanyrespectsthischap-terfollowsthesamelinesasthetreatmentofsecondorderlinearequationsinChapter3.7.1IntroductionSystemsofsimultaneousordinarydifferentialequationsarisenaturallyinproblemsinvolvingseveral
4、dependentvariables,eachofwhichisafunctionofthesamesingleindependentvariable.Wewilldenotetheindependentvariablebytandwillletx1,x2,x3,...representdependentvariablesthatarefunctionsoft.Differentiationwithrespecttotwillbedenotedbyaprime.Forexample,considerthespring–masssyste
5、minFigure7.1.1.ThetwomassesmoveonafrictionlesssurfaceundertheinfluenceofexternalforcesF1(t)andF2(t),andtheyarealsoconstrainedbythethreespringswhoseconstantsarek1,k2,andk3,359August7,201221:04c07Sheetnumber2Pagenumber360cyanblack360Chapter7.SystemsofFirstOrderLinearEquatio
6、nsrespectively.UsingargumentssimilartothoseinSection3.7,wefindthefollowingequationsforthecoordinatesx1andx2ofthetwomasses:d2x1m12=k2(x2−x1)−k1x1+F1(t)dt=−(k1+k2)x1+k2x2+F1(t),(1)d2x2m22=−k3x2−k2(x2−x1)+F2(t)dt=k2x1−(k2+k3)x2+F2(t).AderivationofEqs.(1)isoutlinedinProblem17
7、.F(t)F(t)12k1k2k3m1m2x1x2FIGURE7.1.1Atwo-mass,three-springsystem.Next,considertheparallelLRCcircuitshowninFigure7.1.2.LetVbethevoltagedropacrossthecapacitorandIthecurrentthroughtheinductor.Then,referringtoSection3.7andtoProblem19ofthissection,wecanshowthatthevoltageandcu
8、rrentaredescribedbythesystemofequationsdIV=,dtL(2)dVIV=−−,dtCRCwhereListheinductance,Cisthecapacitance,
