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Mathematics (417)
MATH201 (10)
Chapter 3

# ch3-p1.pdf

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School
Department
Mathematics
Course
MATH201
Professor
Callum Quigley
Semester
Winter

Description
Chapter3SecondOrderEquations31IntroductionAlinearsecondorderdierentialequationcanbewrittenintheform2dydyaxaxaxybx2102dxdxForthemostpartwewillassumethatthefunctionsaxaxaxandbxarecontinuousonsome012intervalIRWewillfurtherassumethatax0onIandwritetheequationinstandard2form2dydyPxQxygx312dxdxThefunctiongxissometimescalledaforcingfunctionRecallthatalinearequationiscalledhomogeneousiftheforcingtermgx0andnonhomogeneousotherwiseItissometimesconvenienttowrite31intheformLygwhereListhedierentialoperatordenedby2ddPxQx32L2dxdxDierentialoperatorsactonfunctionstogiveotherfunctionsLetDLdenotethedomainofLThenforLasdenedin32wehaveDLtwicedierentiablefunctions18Math20131INTRODUCTION192dExample31LetL1Then2dx2d77577xx42xxLx2dx2d2x2x2x2x2x2xLeee4ee5e2dx2dcosxcosxcosxcosx0Lcosx2dx2d1Llnxlnxlnxlnx22dxxDenition32AnoperatorLdierentialorotherwiseiscalledlinearifitsatises1LLLforallDL1212122LccLforallDLcRExample33ConsiderthedierentialoperatorsdenedasfollowsLyyyandTyysinyTherstislinearsinceLLL121212121212LcccccLOntheotherhandTisnotlinearsinceTccsinccsincTLemma34ThedierentialoperatorLdenedineq32islinearProofLetDLandletcRbearbitraryThen12LPxQx12121212PxQxPxQx121122LL12andLccPxcQxccPxQxcLThusLislinearRemarkLineardierentialoperatorsactingondierentiablefunctionsisanalogoustomatricesactingonvectorsLlinearoperatorcorrespondstoAmatrixxdierentiablefunctioncorrespondstoxvectorLgdierentialequationcorrespondstoAxbmatrixequation
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