higher order linear equations

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1、August7,201221:04C04Sheetnumber1Pagenumber221cyanblackCHAPTER4HigherOrderLinearEquationsThetheoreticalstructureandmethodsofsolutiondevelopedintheprecedingchapterforsecondorderlinearequationsextenddirectlytolinearequationsofthirdandhigherorder.Inthisc

2、hapterwebrieflyreviewthisgeneralization,takingparticularnoteofthoseinstanceswherenewphenomenamayappear,becauseofthegreatervarietyofsituationsthatcanoccurforequationsofhigherorder.4.1GeneralTheoryofnthOrderLinearEquationsAnnthorderlineardifferentialequ

3、ationisanequationoftheformdnydn−1ydyP0(t)dtn+P1(t)dtn−1+···+Pn−1(t)dt+Pn(t)y=G(t).(1)WeassumethatthefunctionsP0,...,Pn,andGarecontinuousreal-valuedfunctionsonsomeintervalI:α

4、dnydn−1ydyL[y]=dtn+p1(t)dtn−1+···+pn−1(t)dt+pn(t)y=g(t).(2)ThelineardifferentialoperatorLoforderndefinedbyEq.(2)issimilartothesecondorderoperatorintroducedinChapter3.ThemathematicaltheoryassociatedwithEq.(2)iscompletelyanalogoustothatforthesecondorder

5、linearequation;forthisreasonwesimplystatetheresultsforthenthorderproblem.Theproofsofmostoftheresultsarealsosimilartothoseforthesecondorderequationandareusuallyleftasexercises.221August7,201221:04C04Sheetnumber2Pagenumber222cyanblack222Chapter4.Higher

6、OrderLinearEquationsSinceEq.(2)involvesthenthderivativeofywithrespecttot,itwill,sotospeak,requirenintegrationstosolveEq.(2).Eachoftheseintegrationsintroducesanarbitraryconstant.Henceweexpectthattoobtainauniquesolutionitisnecessarytospecifyninitialcon

7、ditionsy(t)=y,y(t)=y,...,y(n−1)(t)=y(n−1),(3)000000wheretmaybeanypointintheintervalIandy,y,...,y(n−1)isanysetofprescribed0000realconstants.Thefollowingtheorem,whichissimilartoTheorem3.2.1,guaranteesthattheinitialvalueproblem(2),(3)hasasolutionandt

8、hatitisunique.Theorem4.1.1Ifthefunctionsp1,p2,...,pn,andgarecontinuousontheopenintervalI,thenthereexistsexactlyonesolutiony=φ(t)ofthedifferentialequation(2)thatalsosatisfiestheinitialconditions(3),wheret0isanypointinI.Thissolutionexiststhroughoutthein