WelcometoMATH226.2x:LinearDifferentialEquations.Thissyllabusprovidesageneraldescriptionofthecoursecontent,theschedule,theassessmentsandgrading,andgeneralguidelines.Pleasecheckthesyllabusifyouhaveanyquestionsregardingtheoperationofthiscourse.

Linear Differential Equations

Phenomenaasdiverseasanautomobilessuspensionsystem,theswayingofabridge,andthedampingofaskyscraperaregovernedbydifferentialequations.MATH226xisanintroductiontothemathematicaltheoryofordinarydifferentialequations.Thiscourseadoptsamoderndynamicalsystemsapproachtothesubject.Thatis,equationsareanalyzedusingqualitative,numerical,andifpossible,symbolictechniques.InMATH226.2x,wewillstudyaspecialclassofdifferentialequationslineardifferentialequationsthatareespeciallyimportant.Manymodelsassumelinearityinordertotakeadvantageoftechniquesthatapplyonlytolinearequations.Wewillanalyzesolutionstolineardifferentialequationswithtwodependentvariablesusingbothanalyticandqualitativetechniques.Wewillalsostudyforcedsecondorderlinearequationsandrelatedphenomenasuchasbeatsandresonance.

About the Team

PaulBlanchardisprofessorofmathematicsatBostonUniversity.HegrewupinSutton,Massachusetts,USA,andspentthreeundergraduateyearsatBrownUniversity.Duringhissenioryear,hedecidedtohaveanadventureandlearnanewlanguage,sohewasanoccasionalstudentattheUniversityofWarwickinEngland.HereceivedhisPh.D.fromYaleUniversity.Hehastaughtmathematicsformorethanthirtyfiveyears,mostatBostonUniversity.HismainareaofmathematicalresearchiscomplexanalyticdynamicalsystemsandtherelatedpointsetsJuliasetsandtheMandelbrotset.HeisaFellowoftheAmericanMathematicalSociety. Formanyofthelasttwentyyears,hiseffortshavefocused

onmodernizingthetraditionalsophomoreleveldifferentialequationscourse.Thateffort

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hasresultedinnumerousworkshopsandminicourses.HehasalsoauthoredfiveeditionsofDifferentialEquationswithRobertL.DevaneyandGlenR.Hall.Whenhebecomesexhaustedfixingtheerrorsmadebyhistwocoauthors,heheadsforthegolfcoursetoenjoyadifferenttypeoffrustration.

PatrickCummingsisaPh.D.candidateintheDepartmentofMathematicsandStatisticsatBostonUniversity.Hisresearchinvolvesextendingthetheoryoffinitedimensionaldynamicalsystemstoinfinitedimensionaldynamicalsystemsdefinedbypartialdifferentialequations.PatrickreceivedhisBachelorofArtsdegreeinMathematicsfromMaristCollegein2012.WhileatBostonUniversity,hehasbeenateachingassistantforMA226,theresidentialequivalentofMATH226x.

Course Outline

Module Content

Module1:LinearSystemsandtheLinearityPrinciple ReleasedonThursday,April30at1:00PMEDT

Linearsystemsareespeciallynicebecausewecompletelyunderstandthestructureofthesetoftheirsolutions.Theyarealsousedtoapproximatenonlineardifferentialequationsincertainsituations.Inthismodule,wediscussanimportantprincipleforlinearsystemsandthestructureofsetofsolutions.

Module2:StraightLineSolutions ReleasedonThursday,May7at1:00PMEDT

Inthismoduleweusethegeometryofthevectorfieldtofindspecialsolutionsoflinearsystems.Thegeometrywillleadustothealgebraicnotionsofeigenvaluesandeigenvectors.

Module3:PhasePortraitsforLinearSystemswithRealEigenvalues ReleasedonThursday,May7at1:00PMEDT

Givenasystemwithtwodistinctrealeigenvalues,wecanusethebehaviorofthestraightlinesolutionsfoundinModule2todeterminethebehaviorofallsolutions.

Module4:ComplexEigenvalues ReleasedonThursday,May14at1:00PMEDT

InModules2&3wesawthat,forsomelinearsystems,certainsolutionslieonstraightlinesinthephaseplane.Unfortunately,thisisnottrueforalllinearsystems.Wewillusethealgebraofeigenvaluesandeigenvectorstodevelopanunderstandingofsystemswithtwocomplexeigenvalues.

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Module5:SpecialCases:RepeatedandZeroEigenvalues ReleasedonThursday,May21at1:00PMEDT

Linearsystemswithonlyoneeigenvalueorazeroeigenvaluearerelativelyrare.Nevertheless,theyarestillimportant.Inthismodule,wewillmodifythemethodsfromthepreviousmodulestohandletheseremainingcases.Afterthismodulewewillbeabletoanalyzealllinearsystemsofdifferentialequations.

MidMOOCQuiz ReleasedonThursday,May21at1:00PMEDT DueonThursday,May28at1:00PMEDT

ThisquizwilltestthetopicspresentedinModules15.Thequizwillbeworth30%ofyouroverallgrade.

Module6:SecondOrderLinearEquations ReleasedonThursday,May28at1:00PMEDT

Thedampedharmonicoscillatoristhesecondorderdifferentialequationthatisoftenusedtomodelphenomenathatbehavelinearly.Themassspringsystemistheclassicexample.Anothercommonexampleisalinearcircuit.Wederiveaguessingtechniquethatappliestothisdifferentialequation.Wealsodiscussthegeometryofthesolutionsthatareobtainedfromthisguessingtechnique.

Module7:TheTraceDeterminantPlane ReleasedonThursday,May28at1:00PMEDT

Modelsthatusedifferentialequationsofteninvolveparameters.Wehaveencounteredanumberofdifferenttypesoflinearsystemsofdifferentialequations.TheTraceDeterminantplaneisawaytovisualizehowthebehaviorofsystemsdependsontheparameters.

Module8:TheForcedHarmonicOscillator ReleasedonThursday,June4at1:00PMEDT

InModule6werevisitedthemassspringsystem.Weanalyzedaphysicalsystemofamassattachedtoaspringthatslidesonatable.Themassissubjecttoarestoringforceprovidedbythespringandtodamping.Inthismodule,weconsidertheeffectofexternalforcessuchastheshakingofthetableorpushingthemass.

Module9:SinusoidalForcing ReleasedonThursday,June4at1:00PMEDT

WestudythedampedforcedharmonicoscillatorfromModule8wheretheexternalforcecanberepresentedasasineorcosinefunction.Thistypeofexternalforceoccursfrequentlyinapplications.Examplesincludetheshakingofabuildingbyanearthquake.

Module10:UndampedForcingandResonance ReleasedonThursday,June11at1:00PMEDT

Wecontinuethestudyofforcedharmonicoscillators.Inthismoduleweconsideranundampedforcedharmonicoscillatorwithsinusoidalforcing.Averydramaticchangeinthequalitativebehaviorofsolutionsoccursasthefrequencyoftheforcingfunctionapproachesthenaturalfrequencyoftheequation.Thisphenomenoniscalledresonance.

FinalExam ReleasedonThursday,June11at1:00PMEDT DueonThursday,June22at1:00PMEDT

Thisexamwilltestalltopicspresentedinthiscourseandwillbeworth50%ofyouroverallgrade.

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EndofCourse Monday,June22at1:00PMEDT

Thecourseofficiallyendsatthistime.Thecontentwillstillbeavailableafterthecoursecloses,butthoseseekingacertificatemustachieveanoverallgradeof50%bythisdate.

Assessments and Grading

Eachmoduleconsistsofaseriesofvideosinterspacedwithbriefexercisesdesignedtohelpyouassessyourunderstandingofthematerialdiscussedinthevideo.Thesecontentcheckexerciseswillbeworth5%ofyouroverallgrade. Attheendofeachmoduletherewillbeanexercisesetthatwillprovidemoredetailedpracticewiththeconceptspresentedinthemodule.Theseexercisesetswillbeworth15%ofyouroverallgrade. TherewillbeamidMOOCquizthatwilltestyourunderstandingoffirstorderlinearsystemsofdifferentialequations.ItwillbereleasedonMay21at1pm(EDT).Toreceivecredit,youmustsubmityouranswersbyMay28at1pm(EDT).Thisexamwillbeworth30%ofyouroverallgrade. ThefinalexamforthecoursewillbereleasedonThursday,June11at1:00pm(EDT).Itwillcoverallofthematerialdiscussedinalltenmodules.Toreceivecredit,youmustsubmityouranswersbyJune22at1pm(EDT).Thefinalexamwillbeworth50%ofyouroverallgrade. WiththeexceptionofthemidMOOCquiz,thedeadlineforallassessmentswillbetheendofthecourse,thatis,June22at1pm(EDT).Youmaydelaycompletionofthecontentcheckexercisesandexercisesetsuntiltheendofthecoursewhilestillgettingcredit.However,westronglyrecommendthatyoucompleteallexercisesasyougo.

Discussion Forum Guidelines Wehopethatyoufindthediscussionforumstobeausefulcomponentofthiscourse.Theyaremeanttobeanareawherethestudentscaninteractwitheachother,askquestions,ortalktothecoursestaff.Wegreatlyencourageyoutousetheseforumsonaregularbasis. Wesupportandencouragetheuseoftheforumtodiscussoraskquestionsaboutexercisesandconsequentlytheirsolutions.Wewillnotdeletequestionsordiscussionsthatcontainsolutionshowever,wedoaskthatyoudonotabusetheforumsasawaytoshareanswerstoexercises. Weaskthatyoudonotpostcommentsthatarederogatory,defamatory,orinanywayattackotherstudents.Becourteousandshowthesamerespectyouhopetoreceive.

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Discussionforummoderatorswilldeletepoststhatarerude,inappropriate,orofftopic.Commenterswhorepeatedlyabusethispublicforumwillberemovedfromthecourse. Thereisafeatureinthediscussionforumsthatallowsyoutoselectfromtwoposttypes,QuestionandDiscussion.TheQuestiontypeismeantforspecificissueswiththeplatformorwithcontent,andtheDiscussiontypeismeanttoshareideasandstartconversation.Pleasekeepthisdistinctioninmindwhenpostingtothediscussionforum.

FAQ Q:ShouldIemailtheprofessororanypersonsinvolvedwiththiscoursedirectly? A:No.Ifyoufeeltheneedtocontactthecoursestaffinvolvedinthiscourse,pleasedosothroughtheDiscussionForum. Q:DoIneedtobuyanypersonalmaterialstotakethiscourse? A:No.Youdonotneedtopurchasetextbooksoranymaterialstoaidyouincompletingthecourse. Q:I'venevertakenanedXcoursebeforeandthisisconfusing.WhatdoIdo? A:ThereisaprecourseedXwalkthroughthatbeginnerscanwatch.ItexplainsindetailhowtousetheedXplatform.Forfurtherinformation,pleasevisitthedemoedXcourse. Q:Ifoundamistakeinthecourse.WheredoIreportit? A:OntheWikipage,thereisaspecificsectionforErrata.Youcangothere,editthepage,andpostinformationconcerninganyerrorsorissuesyouhavefound.Wewilltrytofixthemassoonaspossible. Q:HowdoIlearnmoreaboutthemathematicsdiscussedinModulex? A:Manyofthemodulesdiscusstopicsthatcanbestudiedinmuchmoredetail.Ifyoufindatopicespeciallyinterestingandwouldliketoknowmore,thenpleasepostaquestiononthediscussionforum.Ifweknowofagoodreferenceorresource,thenwewillpostitonthewiki.

Time Zones Anoteabouttimereferences:TimewillbereportedbycoursestaffasEasternDaylightTime,NorthAmerica(EDT).AnytimeslistedbyedX,suchasduedateslistedonthecoursesite,willbereportedinUniversalTimeCoordinated(UTC).Thecoursestaffwillmakeeveryefforttomaketimesandtimezonesasclearaspossible.Therearevarioustimezoneconvertersonthewebsuchashttp://www.timeanddate.com/worldclock/converter.html.

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Honor Code TheedXplatformassumesacertainlevelofdecorumandresponsibilityfromthosetakingthiscourse.PleasereviewtheedXHonorCode,whichisreproducedbelow. ByenrollinginanedXcourse,IagreethatIwill:

Completeallmidtermsandfinalexamswithmyownworkandonlymyownwork.Iwillnotsubmittheworkofanyotherperson.

Maintainonlyoneuseraccountandnotletanyoneelseusemyusernameand/orpassword.

Notengageinanyactivitythatwoulddishonestlyimprovemyresults,orimproveorhurttheresultsofothers.

Notpostanswerstoproblemsthatarebeingusedtoassessstudentperformance. UnlessotherwiseindicatedbytheinstructorofanedXcourse,learnersonedXareencouragedto:

Collaboratewithothersonthelecturevideos,exercises,homeworkandlabs. Discusswithothersgeneralconceptsandmaterialsineachcourse. PresentideasandwrittenworktofellowedXlearnersorothersforcommentor

criticism.

Credits and Acknowledgements Aswithanymajoreffort,thiscoursewouldnotbepossiblewithoutlargecontributionsfrommanysources.Wewouldliketoextendaspecialthankstothevariousteamswhohaveputinuncountablehoursofworktohelpcreatethiscourse.Specifically,wewanttothankthefollowingpeopleandorganizationsthathavecontributedalargeamountofefforttomakethiscoursebecomeareality:RomyRuukel,TimBrenner,VanessaRuanoforadministratingthisprocessandbeingresponsibleforeveryaspectofmakingthiscourseJoeDwyerforeditingtheannotatedslidevideosthatappearinthiscourseKellanReckforfilmingandeditingtheaboutvideoCourtneyTeixeirawhodrewtheimagesonthetitlecardsAndrewAbrahamsonandAdamBrillaofBUsMetropolitanCollegewhohelpeduswithourtabletcaptureintheirmediaroomDanielShankforaccuracycheckingProfessorJohnPolkingofRiceUniversityforlettingususehisprogrampplaneinthiscourseMathWorksforprovidinglicensesforMATLABduringthecourseJohnKotwicki,BrandonArmstrong,andespeciallyErinByrneofMathWorksfortheirassistancewithMATLABHubertHohnwhoworkedwithusdesigningandimplementingDETools,softwarethatweusewhenweteachdifferentialequationsCengageLearningforprovidingpartialsupportduringthedevelopmentofDEToolsandtheDigitalLearningInitiativeandtheDepartmentofMathematicsandStatisticsatBostonUniversityforsupportingPaulBlanchardandPatrickCummingsduringthedevelopmentofthiscourse.

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ThiscoursewouldnothavebeenpossibleiftheNationalScienceFoundationhadnotpartiallyfundedtheBostonUniversityDifferentialEquationsProjectfrom1993to1998. ManyundergraduateandgraduatestudentshaveworkedontheBUDifferentialEquationsProjectovertheyears:GarethRoberts,AlexKasman,BrianPersaud,MelissaVellela,SamKaplan,BillBasener,SebastianMarotta,StephanieR.Jones,AdrianVajiac,DanielCuzzocreo,DuffCampbell,LeeDeville,J.DougWright,DanLook,NuriaFagella,NickBenes,AdrianIovita,KinyaOno,andBeverlySteinhoff. PaulBlanchardwouldespeciallyliketothankhiscolleaguesandcoauthors,RobertL.DevaneyandGlenR.Hall,formanyyearsofenjoyablecollaborationonthedevelopmentofmaterialsusedtoteachdifferentialequations.

Terms of Service Forfurtherinformation,pleasereviewtheedXTermsofService(https://www.edx.org/edxtermsservice).

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