Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Forfurthervolumes: http://www.springer.com/series/666 William A. Adkins • Mark G. Davidson Ordinary Differential Equations 123 WilliamA.Adkins MarkG.Davidson DepartmentofMathematics DepartmentofMathematics LouisianaStateUniversity LouisianaStateUniversity BatonRouge,LA BatonRouge,LA USA USA ISSN0172-6056 ISBN978-1-4614-3617-1 ISBN978-1-4614-3618-8(eBook) DOI10.1007/978-1-4614-3618-8 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012937994 MathematicsSubjectClassification(2010):34-01 ©SpringerScience+BusinessMediaNewYork2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface This text is intended for the introductorythree- or four-hourone-semester sopho- more level differential equations course traditionally taken by students majoring in science or engineering. The prerequisite is the standard course in elementary calculus. EngineeringstudentsfrequentlytakeacourseonandusetheLaplacetransform asanessentialtoolintheirstudies.Inmostdifferentialequationstexts,theLaplace transformispresented,usuallytowardtheendofthetext,asanalternativemethod forthesolutionofconstantcoefficientlineardifferentialequations,withparticular emphasisondiscontinuousorimpulsiveforcingfunctions.Becauseofitsplacement at the end of the course, this important concept is not as fully assimilated as one mighthopeforcontinuedapplicationsintheengineeringcurriculum.Thus,agoal of the present text is to present the Laplace transform early in the text, and use it asatoolformotivatinganddevelopingmuchoftheremainingdifferentialequation conceptsforwhichitisparticularlywellsuited. There are severalrewardsfor investingin an early developmentof the Laplace transform.Thestandardsolutionmethodsforconstantcoefficientlineardifferential equations are immediate and simplified. We are able to provide a proof of the existenceanduniquenesstheoremswhicharenotusuallygiveninintroductorytexts. Thesolutionmethodforconstantcoefficientlinearsystemsisstreamlined,andwe avoidhavingtointroducethenotionofadefectiveornondefectivematrixordevelop generalizedeigenvectors.EventheCayley–Hamiltontheorem,usedinSect.9.6,is a simple consequenceofthe Laplace transform.Inshort, the Laplacetransformis aneffectivetoolwithsurprisinglydiverseapplications. Mathematicians are well aware of the importance of transform methods to simplify mathematical problems.For example, the Fourier transform is extremely important and has extensive use in more advanced mathematics courses. The wavelet transform has received much attention from both engineers and mathe- maticiansrecently.It has been appliedto problemsin signal analysis, storage and transmission of data, and data compression. We believe that students should be introducedtotransformmethodsearlyonintheirstudiesandtothatend,theLaplace transform is particularly well suited for a sophomore level course in differential v vi Preface equations. It has been our experience that by introducing the Laplace transform nearthebeginningofthetext,studentsbecomeproficientinitsuseandcomfortable withthisimportantconcept,whileatthesametimelearningthestandardtopicsin differentialequations. Chapter1isaconventionalintroductorychapterthatincludessolutiontechniques forthemostcommonlyusedfirstorderdifferentialequations,namely,separableand linearequations,andsomesubstitutionsthatreduceotherequationstooneofthese. TherearealsothePicardapproximationalgorithmandadescription,withoutproof, ofanexistenceanduniquenesstheoremforfirstorderequations. Chapter 2 starts immediatelywith the introductionof the Laplace transformas anintegraloperatorthatturnsadifferentialequationint intoanalgebraicequation inanothervariables.Afewbasiccalculationsthenallowonetostartsolvingsome differential equations of order greater than one. The rest of this chapter develops the necessary theory to be able to efficiently use the Laplace transform. Some proofs,suchastheinjectivityoftheLaplacetransform,aredelegatedtoanappendix. Sections2.6and2.7introducethebasicfunctionspacesthatareusedtodescribethe solutionspacesofconstantcoefficientlinearhomogeneousdifferentialequations. WiththeLaplacetransforminhand,Chap.3efficientlydevelopsthebasictheory for constant coefficient linear differential equations of order 2. For example, the homogeneous equation q.D/y D 0 has the solution space E that has already q been describedin Sect.2.6. The Laplace transformimmediately givesa very easy procedureforfindingthetestfunctionwhenteachingthemethodofundetermined coefficients. Thus, it is unnecessary to develop a rule-based procedure or the annihilatormethodthatiscommoninmanytexts. Chapter 4 extends the basic theory developed in Chap.3 to higher order equations.Allofthebasicconceptsandproceduresnaturallyextend.Ifdesired,one cansimultaneouslyintroducethehigherorderequationsasChap.3isdevelopedor verybrieflymentionthedifferencesfollowingChap.3. Chapter5introducessomeofthetheoryforsecondorderlineardifferentialequa- tionsthatarenotconstantcoefficient.Reductionoforderandvariationofparameters are topics that are included here, while Sect.5.4 uses the Laplace transform to transformcertainsecondordernonconstantcoefficientlineardifferentialequations intofirstorderlineardifferentialequationsthatcanthenbesolvedbythetechniques describedinChap.1. We have broken up the main theory of the Laplace transform into two parts for simplicity.Thus, the materialin Chap.2 onlyuses continuousinputfunctions, while in Chap.6 we return to develop the theory of the Laplace transform for discontinuousfunctions,mostnotably,the step functionsandfunctionswith jump discontinuities that can be expressed in terms of step functions in a natural way. The Dirac delta function and differential equationsthat use the delta function are also developedhere. The Laplace transformworks very well as a tool for solving such differential equations. Sections 6.6–6.8 are a rather extensive treatment of periodic functions, their Laplace transform theory, and constant coefficient linear differentialequationswithperiodicinputfunction.Thesesectionsmakeforagood supplementalprojectforamotivatedstudent. Preface vii Chapter 7 is an introduction to power series methods for linear differential equations.AsaniceapplicationoftheFrobeniusmethod,explicitLaplaceinversion formulas involving rational functions with denominators that are powers of an irreduciblequadraticarederived. Chapter8isprimarilyincludedforcompleteness.Itisastandardintroductionto somematrixalgebrathatisneededforsystemsoflineardifferentialequations.For thosewhohavealreadyhadexposuretothisbasicalgebra,itcanbesafelyskipped orgivenassupplementalreading. Chapter 9 is concerned with solving systems of linear differential equations. By the use of the Laplace transform˚, it is possib(cid:2)le to give an explicit formula for the matrix exponential eAt D L(cid:2)1 .sI (cid:2)A/(cid:2)1 that does not involve the use of eigenvectors or generalized eigenvectors. Moreover, we are then able to develop an efficient method for computing eAt known as Fulmer’s method. Another thing whichissomewhatuniqueisthatweusethematrixexponentialinordertosolvea constantcoefficientsystemy0 DAyCf.t/,y.t /Dy bymeansofanintegrating 0 0 factor.Animmediateconsequenceofthisistheexistenceanduniquenesstheorem for higherorderconstantcoefficientlinear differentialequations,a factthat is not commonlyprovedintextsatthislevel. Thetexthasnumerousexercises,withanswerstomostodd-numberedexercises intheappendix.Additionally,astudentsolutionsmanualisavailablewithsolutions to most odd-numbered problems, and an instructors solution manual includes solutionstomostexercises. Chapter Dependence Thefollowingdiagramillustratesinterdependenceamongthechapters. 1 2 3 8 4 5 6 9 7 viii Preface Suggested Syllabi Thefollowingtablesuggeststwopossiblesyllabiforonesemestercourses. 3-HourCourse 4-HourCourse FurtherReading Sections1.1–1.6 Sections1.1–1.7 Sections2.1–2.8 Sections2.1–2.8 Sections3.1–3.6 Sections3.1–3.7 Sections4.1–4.3 Sections4.1–4.4 Section4.5 Sections5.1–5.3,5.6 Sections5.1–5.6 Sections6.1–6.5 Sections6.1–6.5 Sections6.6–6.8 Sections7.1–7.3 Section7.4 Sections9.1–9.5 Sections9.1–9.5,9.7 Section9.6 SectionsA.1,A.5 Chapter 8 is on matrix operations. It is not included in the syllabi given above sincesomeofthismaterialissometimescoveredbycoursesthatprecededifferential equations. Instructors should decide what material needs to be covered for their students.ThesectionsintheFurtherReadingcolumnarewrittenatamoreadvanced level.Theymaybeusedtochallengeexceptionalstudents. We routinely provide a basic table of Laplace transforms, such as Tables 2.6 and2.7,forusebystudentsduringexams. Acknowledgments We would like to express our gratitude to the many people who have helped to bring this text to its finish. We thank Frank Neubrander who suggested making the Laplace transformhave a more central role in the developmentof the subject. We thank the many instructors who used preliminary versions of the text and gave valuable suggestions for its improvement. They include Yuri Antipov, Scott Baldridge, Blaise Bourdin, Guoli Ding, Charles Egedy, Hui Kuo, Robert Lipton, Michael Malisoff, Phuc Nguyen, Richard Oberlin, Gestur Olafsson, Boris Rubin, Li-Yeng Sung, Michael Tom, Terrie White, and Shijun Zheng. We thank Thomas Davidsonforproofreadingmanyofthesolutions.Finally,wethankthemanymany studentswhopatientlyusedversionsofthetextduringitsdevelopment. BatonRouge,Louisiana WilliamA.Adkins MarkG.Davidson Contents 1 FirstOrderDifferentialEquations......................................... 1 1.1 AnIntroductiontoDifferentialEquations............................. 1 1.2 DirectionFields......................................................... 17 1.3 SeparableDifferentialEquations ...................................... 27 1.4 LinearFirstOrderEquations........................................... 45 1.5 Substitutions............................................................ 63 1.6 ExactEquations ........................................................ 73 1.7 ExistenceandUniquenessTheorems.................................. 85 2 TheLaplaceTransform ..................................................... 101 2.1 LaplaceTransformMethod:Introduction............................. 101 2.2 Definitions,BasicFormulas,andPrinciples .......................... 111 2.3 PartialFractions:ARecursiveAlgorithmforLinearTerms.......... 129 2.4 PartialFractions:ARecursiveAlgorithmforIrreducible Quadratics............................................................... 143 2.5 LaplaceInversion....................................................... 151 2.6 TheLinearSpacesE :SpecialCases.................................. 167 q 2.7 TheLinearSpacesE :TheGeneralCase ............................. 179 q 2.8 Convolution............................................................. 187 2.9 SummaryofLaplaceTransformsandConvolutions.................. 199 3 SecondOrderConstantCoefficientLinearDifferentialEquations.... 203 3.1 Notation,Definitions,andSomeBasicResults....................... 205 3.2 LinearIndependence................................................... 217 3.3 LinearHomogeneousDifferentialEquations ......................... 229 3.4 TheMethodofUndeterminedCoefficients ........................... 237 3.5 TheIncompletePartialFractionMethod.............................. 245 3.6 SpringSystems......................................................... 253 3.7 RCLCircuits............................................................ 267 ix