second order linear equations

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1、August7,201221:04c03Sheetnumber1Pagenumber137cyanblackCHAPTER3SecondOrderLinearEquationsLinearequationsofsecondorderareofcrucialimportanceinthestudyofdifferentialequationsfortwomainreasons.Thefirstisthatlinearequationshavearichtheo-reticalstructurethatunderlie

2、sanumberofsystematicmethodsofsolution.Further,asubstantialportionofthisstructureandofthesemethodsisunderstandableatafairlyelementarymathematicallevel.Inordertopresentthekeyideasinthesim-plestpossiblecontext,wedescribetheminthischapterforsecondorderequations.A

3、notherreasontostudysecondorderlinearequationsisthattheyarevitaltoanyseriousinvestigationoftheclassicalareasofmathematicalphysics.Onecannotgoveryfarinthedevelopmentoffluidmechanics,heatconduction,wavemotion,orelectromagneticphenomenawithoutfindingitnecessarytoso

4、lvesecondorderlin-eardifferentialequations.Asanexample,wediscusstheoscillationsofsomebasicmechanicalandelectricalsystemsattheendofthechapter.3.1HomogeneousEquationswithConstantCoefficientsAsecondorderordinarydifferentialequationhastheformd2ydy=ft,y,,(1)dt2dtw

5、herefissomegivenfunction.Usually,wewilldenotetheindependentvariablebytsincetimeisoftentheindependentvariableinphysicalproblems,butsome-timeswewillusexinstead.Wewillusey,oroccasionallysomeotherletter,todesignatethedependentvariable.Equation(1)issaidtobelineari

6、fthefunctionf137August7,201221:04c03Sheetnumber2Pagenumber138cyanblack138Chapter3.SecondOrderLinearEquationshastheformdydyft,y,=g(t)−p(t)−q(t)y,(2)dtdtthatis,iffislinearinyanddy/dt.InEq.(2)g,p,andqarespecifiedfunctionsoftheindependentvariabletbutdonotdependon

7、y.InthiscaseweusuallyrewriteEq.(1)asy+p(t)y+q(t)y=g(t),(3)wheretheprimesdenotedifferentiationwithrespecttot.InsteadofEq.(3),weoftenseetheequationP(t)y+Q(t)y+R(t)y=G(t).(4)Ofcourse,ifP(t)=0,wecandivideEq.(4)byP(t)andtherebyobtainEq.(3)withQ(t)R(t)G(t)p(

8、t)=,q(t)=,g(t)=.(5)P(t)P(t)P(t)IndiscussingEq.(3)andintryingtosolveit,wewillrestrictourselvestointervalsinwhichp,q,andgarecontinuousfunctions.1IfEq.(1)isnotoftheform(3)or(4),thenitiscalle