Part1OrdinaryDifferentialEquations(常微分方程式微分变数只有一个)

--#-FIGURE2.11Resonance.FIGURE2.11Resonance.TheTacomaNarrowsBridgewascompletedin1940andstoodasanewstandardofcombinedartistryandfunctioncility^Thebridgesoonbecameknownforitstemlcncytoinhighwinds,butnoonesuspectedwhatwasabouttooccur.OnNovember7>1940、energyprovidedbyunusuallystrongwinds,coupledwitharesonatingeffectinthebridge\matenalanddesig/itcausedtheCfSclUatlonsinthebridgetobereinforedandbuildiodangerouslevels,Sno叫thetwistingcausedonesideofikesidewalkiqrise28feetaboveihotoftheotherside.Concretedroppedoutoftheroadway,andasectionofthe^pensionspancon9耘蹿［yrotatedandfellaway.Shortlythereaftertheentirecenter印即collapsedinfoPugetSoitmLThissensationalccm^tructUmfailuremotivatednewmathematicaltreati?ienfsofvibrationand1vqyephenomenciinthedesignofbridgesandotherlargestructures.Theforcesthatbroughtdownthisbridgeareamorecomplicatedversion(^'ther^^onancephenomenondiscussedinSection2.7.3,

2.10MethodofVariationofParameters( 燮换法),Supposewecanfindafundamentalsetofsolutionsy1andy2forthehomogeneousequation.Tryy=uy+vy（?斯吴法（trial-and-errormethod）：找到期限多麻特解中的任何一雄）12一，2.10MethodofVariationofParameters( 燮换法),Supposewecanfindafundamentalsetofsolutionsy1andy2forthehomogeneousequation.Tryy=uy+vy（?斯吴法（trial-and-errormethod）：找到期限多麻特解中的任何一雄）12一，ypletu'y+v'y=012y'=uy'+vy'p 12(1)y""u'y'+uy"+v'y'+yy"1122Substituteintothenon-homogeneousequationu'y'+uy"+v'y'+vy"+p(uy'11221+vy')+q(uy+vy)=f(%)2 12u(y"+py'+qy)+v(y"+py'+qy)+u'y'+v'y'=f(%)1 112 22u'y'+v'y'=f(%)（2）（1）y'-(2)y2nu'(y1y2'-y2y1')=-y2f(%)(消去v')yf(%)

2 yy'-y'y12 12(1)y'(1)y'-(2)yn11v'(y2y1'-y1y2')=-yfG)(消去v'yf(%)

1 v'yy'-y'y12 12Foranynon-vanishingpairofuandv,wecanusethemtofindy.p，定羲——TheWronskianofthetwosolutionsyandyisdefinedasy(%)1y(%)1yf(%)1y(%)2y'(%)2=y(%),y'(%)-y'(%),y(%)1212WronskianTest—Thetwosolutionsyandy12

arelinearlydependentonIifandonlyifW(x)=0forallxinI.Example2.15Wewillfindthegeneralsolutionofyff+4y=sec(x)for一洋/4<x<ji/4.Thecharacteristicequationofy"4-4y=0isA2+4=0,withroots±2LWemaythereforechoose(耳)=cos(2x)and尤)=sin(2x)*TheWronskianofthesesolutionsofthehomogeneousequationis8s(2龙)sin(2x)™2sin(2%)2cos(28s(2龙)sin(2x)™2sin(2%)2cos(21)=2.With/(x)=sec(x),equations(2.12)giveusuf(x)= sm(2x)第c(x)=—：2sin(^)cos(x)——―一sin(^)2 cos(%)andvf(x)=：cos(2r)secQ)=-[2cus2(x)-1]■2 2 cos(x)=cos(jt)-；sec(jt).Thenandm(x)=—sin(^)m(x)=—sin(^)dx=cos(^)vU)=jcos(x)dx-/sec(x)dx

=sin(x)-；ln|sec(x)+tan(x)|.Herewehavelettheconstantsofintegrationbezerobecauseweneedonlyoneuandonev.Nowwehavetheparticularsolution忤(芯)=mU)y]U)+v(x)y2(x)=cos(%)cos(2x)+^sin(x)—；ln|secU)+sin(2x).Thegeneralsolutionofyif+4y=sec(x)isyM=yhW4- =c\cos(2^)+c±sin(2x)+cos(%)cos(2%)+fsin(x)—injsec(jc)+tan(x)Asin(2x).瞬Example2.16Supposewewantthegeneralsolutionofyf,—-yf+[y=x2+\forx>0.Theassociatedhomogeneousequationisyft--y?+—^y=0,whichwerecognizeasanEulerequation,withfundamentalsolutionsyi(x)=xand”=x4forjc>0.TheWronskianofthesesolutionsisandthisisnonzeroforx>andIntegratetogetandAparticularsoiutionis力。Thegenera!solutionisyM