Springer Undergraduate Mathematics Series Robert Magnus Essential Ordinary Differential Equations Springer Undergraduate Mathematics Series AdvisoryEditors MarkA.J.Chaplain,St.Andrews,UK AngusMacintyre,Edinburgh,UK SimonScott,London,UK NicoleSnashall,Leicester,UK EndreSüli,Oxford,UK MichaelR.Tehranchi,Cambridge,UK JohnF.Toland,Bath,UK The Springer UndergraduateMathematics Series (SUMS) is a series designed for undergraduatesinmathematicsandthesciencesworldwide.Fromcorefoundational material to final year topics, SUMS books take a fresh and modern approach. Textual explanations are supported by a wealth of examples, problems and fully- workedsolutions,withparticularattentionpaidtouniversalareasofdifficulty.These practicaland concise texts are designed for a one- or two-semester course but the self-studyapproachmakesthemidealforindependentuse. Robert Magnus Essential Ordinary Differential Equations RobertMagnus MathematicsDivision UniversityofIceland Reykjavik,Iceland ISSN1615-2085 ISSN2197-4144 (electronic) SpringerUndergraduateMathematicsSeries ISBN978-3-031-11530-1 ISBN978-3-031-11531-8 (eBook) https://doi.org/10.1007/978-3-031-11531-8 Mathematics Subject Classification: 34-01, 34A12, 34A30, 34B09, 34B24, 34B27, 34B05, 34B24, 34B27 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To thememoryof AlbertandGertrudeMagnus Preface The title of this book indicates a personal view concerning what topics should be included under the heading of “essential.” I am sure that many readers will disagreewithsomeofmychoices.Isitreallyessentialtoincludesomuchexistence theory? Is Sturm-Liouville theory essential? Should not some numerical methods beincluded?Andwhataboutsomequalitativestudiesofplanesystems? Whatguidedmewasasimplethought.Manyundergraduatestudentsofmathe- maticswillstudydifferentialequationstoalimitedextentonly.Whichtopicscould be viewed as necessary knowledge to serve up in a course for undergraduates, bearing in mind that differential equations pervade the whole of physics and engineering,aswellasmanyalliedbranchesofmathematics?Weshouldalsobearin mindthatthosewishingtolearnmorewilldoubtlessbeofferedavarietyofoptional courses, goingbeyondthe essential topics, and tailored to the specific interests of thestudent.Somewillincludetheoreticalstudiesofdynamicalsystems;otherswill include computationalmathematics. While other students will have no interest in pursuingdifferentialequationsanyfurther,butshouldsurely comeaway knowing somethinguseful. The topics presented here could form the basis of such a course. Most of the material should be accessible to a student in their second (or certainly third) undergraduate year. Throughout the text we have mathematics students in mind. Therefore, a high standard of rigour is maintained where possible. Moreover, essential does not mean simple or elementary. The topics selected are treated in some depth. Demands are made of the reader from the start. In the opening paragraph,weplungestraightintothetheoryoflineardifferentialequationsoforder n, without tiptoeing through the second order case first. From Chap.5 onwards, we probablygoa bitbeyondthe essentialand the materialbecomesin partsquite challenging,butanyteacherwillexercisechoiceoverwhattoomit. There is absolutely no reason why a text of this kind should try to be self- contained,anddevelopfromscratchallmethodsandconceptsused.Nocourseexists in its own vacuum. It is assumed that the reader is acquainted with multivariate calculus, for example partial derivatives, the chain rule, multiple integrals, and understands concepts of rigorous analysis, such as limits, function series and vii viii Preface uniformconvergence.Linearalgebra (vectorspacesand matrices) is used without hesitation.Occasionallycomplexanalysisisneeded,andis especiallyrelied onin Chap.7.TheLebesgueintegralisavoided,whichcausessomeawkwardness,again, especiallyinChap.7.Alittlemetricspacetheoryisoccasionallyneeded,butmainly thingsthatthereadermayhaveacquiredinacourseofmultivariatecalculus,suchas theBolzano-WeierstrasstheoremandthenotionofacompactsetinRn.InChap.7, itcouldhelpifthe readerunderstandsthatthereare differentconvergencenotions forfunctionseries.Again,inconnectionwithChap.7,itwouldbeniceifthestudent wasconcurrentlylearningsomeclassicalFourierseries.Broadlyspeaking,werely on the mathematicsthat the student has learned in their first year and is currently learningintheirsecondorthirdyear.Chapters6and7makethegreatestdemands onthereader’sknowledgeofanalysisandmetricspaces,whilstChaps.5and6make thegreatestdemandsforknowledgeoflinearalgebra. A large number of exercises is included, some with hints. Many are quite challenging. There are also sections presenting so-called “projects,” (for want of a better term). In them, text and exercises alternate; the latter are enumerated by Arabicnumeralsprefixedbyaletter“A,”“B,”etc.,tomakeforeasyreference.The projectsgoabitbeyondtheessential,andifthisworkisusedasateachingtext,the newideasthatareintroducedcanbestudiedbystudentsontheirown,oringroups, andbethebasisofstudentpresentations.Alistofalltheprojectscanbeperusedby consulting“Projects”intheindexatthebackofthistext. Reykjavik,Iceland RobertMagnus April2022 Contents 1 LinearOrdinaryDifferentialEquations .................................. 1 1.1 FirstOrderLinearEquations........................................... 2 1.2 ThenthOrderLinearEquation ........................................ 3 1.2.1 TheWronskian ................................................. 5 1.2.2 Non-homogeneousEquations ................................. 6 1.2.3 ComplexSolutions ............................................. 9 1.2.4 Exercises ........................................................ 10 1.2.5 Projects ......................................................... 15 1.3 HomogeneousLinearEquationswithConstantCoefficients ......... 18 1.3.1 WhattodoAboutMultipleRoots ............................. 20 1.3.2 Euler’sEquation ................................................ 22 1.3.3 Exercises ........................................................ 24 1.4 Non-homogeneousEquationswithConstantCoefficients ........... 25 1.4.1 HowtoCalculateaParticularSolution ....................... 27 1.4.2 Exercises ........................................................ 32 1.4.3 Projects ......................................................... 34 1.5 BoundaryValueProblems ............................................. 37 1.5.1 BoundaryConditions .......................................... 41 1.5.2 Green’sFunction ............................................... 42 1.5.3 Exercises ........................................................ 47 2 SeparationofVariables...................................................... 53 2.1 SeparableEquations .................................................... 53 2.1.1 TheAutonomousCase ......................................... 54 2.1.2 TheNon-autonomousCase .................................... 60 2.1.3 Exercises ........................................................ 62 2.2 One-ParameterGroupsofSymmetries ................................ 65 2.2.1 Exercises ........................................................ 70 2.3 Newton’sEquation ..................................................... 71 2.3.1 MotioninaRegularLevelSet ................................. 73 2.3.2 CriticalPoints .................................................. 76 ix x Contents 2.3.3 Exercises ........................................................ 79 2.4 MotioninaCentralForceField ....................................... 83 3 SeriesSolutionsofLinearEquations ...................................... 91 3.1 SolutionsatanOrdinaryPoint ......................................... 91 3.1.1 PreliminariesonPowerSeries ................................. 91 3.1.2 SolutioninPowerSeriesatanOrdinaryPoint ............... 93 3.1.3 Exercises ........................................................ 99 3.1.4 Projects ......................................................... 99 3.2 SolutionsataRegularSingularPoint ................................. 101 3.2.1 TheMethodofFrobenius ...................................... 102 3.2.2 TheSecondSolutionWhenγ −γ IsanInteger ............ 109 1 2 3.2.3 ThePointatInfinity ............................................ 112 3.2.4 Exercises ........................................................ 115 3.2.5 Projects ......................................................... 116 4 ExistenceTheory............................................................. 121 4.1 ExistenceandUniquenessofSolutions ............................... 121 4.1.1 Picard’sTheoremandSuccessiveApproximations .......... 123 4.1.2 ThenthOrderLinearEquationRevisited ..................... 132 4.1.3 TheFirstOrderVectorEquation .............................. 134 4.1.4 Exercises ........................................................ 137 4.1.5 Projects ......................................................... 139 5 TheExponentialofaMatrix................................................ 147 5.1 DefiningtheExponential............................................... 147 5.1.1 Exercises ........................................................ 149 5.2 CalculationofMatrixExponentials ................................... 150 5.2.1 EigenvectorMethod ............................................ 150 5.2.2 Cayley-Hamilton ............................................... 154 5.2.3 InterpolationPolynomials ..................................... 156 5.2.4 Newton’sDividedDifferences ................................ 157 5.2.5 AnalyticFunctionsofaMatrix ................................ 161 5.2.6 Exercises ........................................................ 164 5.2.7 Projects ......................................................... 169 5.3 LinearSystemswithVariableCoefficients ............................ 171 5.3.1 Exercises ........................................................ 175 5.3.2 Projects ......................................................... 177 6 ContinuationofSolutions................................................... 183 6.1 TheMaximalSolution ................................................. 183 6.1.1 Exercises ........................................................ 188 6.2 DependenceonInitialConditions ..................................... 190 6.2.1 Differentiabilityofφx ......................................... 195 x0 6.2.2 HigherDerivativesofφx ...................................... 200 x0 6.2.3 EquationswithParameters .................................... 205 6.2.4 Exercises ........................................................ 207