applications of higher-order differential equations

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1、ApplicationsofHigher-OrderDifferential5EquationsInChapter4,wediscussedseveraltechniquesforsolvinghigher-orderdifferentialequations.Inthischapter,weillustratehowsomeofthesemethodscanbeusedtosolveinitial-valueproblemsthatmodelphysicalsituations.5.1Harmo

2、nicMotion5.1.1SimpleHarmonicMotionSupposethatamassisattachedtoanelasticspringthatissuspendedfromarigidsupportsuchasaceiling.AccordingtoHooke’slaw,thespringexertsarestoringforceintheupwarddirectionthatisproportionaltothedisplacementofthespring.Hooke’sL

3、aw:Fks,wherek>0istheconstantofproportional-ityorspringconstant,andsisthedisplacementofthespring.Aspringhasnaturallengthb.Whenamassisattachedtothespring,itisstretchedsunitspastitsnaturallengthtotheequilibriumpositionx0.Whenthesystemisputintomotion,th

4、edisplacementfromx0attimetisgivenbyxt.ByNewton’sSecondLawofMotion,Fmamd2x/dt2,wheremrepresentsmassandarepresentsacceleration.Ifweassumethattherearenootherforces321322Chapter5ApplicationsofHigher-OrderDifferentialEquationsactingonthemass,thenwedet

5、erminethedifferentialequationthatmodelsthissituationinthefollowingway:d2xmforcesactingonthesystemdt2ksxmgkskxmg.Atequilibriumksmg,soaftersimplification,weobtainthedifferentialequationd2xd2xmkxormkx0.dt2dt2Thetwoinitialconditionsthata

6、reusedwiththisproblemaretheinitialdisplace-mentx0Αandinitialvelocitydx/dt0Β.Hence,thefunctionxtthatdescribesthedisplacementofthemasswithrespecttotheequilibriumpositionisfoundbysolvingtheinitial-valueproblemd2xmkx0dt2(5.1)dxx0Α,0Β.dtTh

7、edifferentialequationininitial-valueproblem(5.1)disregardsallretardingforcesactingonthemotionofthemass.Thesolutionxttothisproblemrepresentsthedisplacementofthemassattimet.Basedontheassumptionsmadeinderivingthedifferentialequation(thepositivedirectio

8、nisdown),positivevaluesofxtindicatethatthemassisbeneaththeequilibriumpositionwhilenegativevaluesofxtindicatethatthemassisabovetheequilibriumposition.EXAMPLE5.1.1:Amassweighing60lbstretchesaspring6inches.Determinethefunctionxtthatdescribe