Springer Undergraduate Texts in Mathematics and Technology Christian Constanda Diff erential Equations A Primer for Scientists and Engineers Second Edition Springer Undergraduate Texts in Mathematics and Technology SeriesEditors: J.M.Borwein,Callaghan,NSW,Australia H.Holden,Trondheim,Norway V.H.Moll,NewOrleans,LA,USA EditorialBoard: L.Goldberg,Berkeley,CA,USA A.Iske,Hamburg,Germany P.E.T.Jorgensen,IowaCity,IA,USA S.M.Robinson,Madison,WI,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/7438 Christian Constanda Differential Equations A Primer for Scientists and Engineers Second Edition 123 ChristianConstanda TheCharlesW.OliphantProfessor ofMathematicalSciences DepartmentofMathematics TheUniversityofTulsa Tulsa,OK,USA ISSN1867-5506 ISSN1867-5514 (electronic) SpringerUndergraduateTextsinMathematicsandTechnology ISBN978-3-319-50223-6 ISBN978-3-319-50224-3 (eBook) DOI10.1007/978-3-319-50224-3 LibraryofCongressControlNumber:2016961538 MathematicsSubjectClassification(2010):34-01 ©SpringerInternationalPublishingAG2013,2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproduction onmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface to the Second Edition The second edition of this book contains a number of changes, some of substance and somemerelycosmetic.Thesechangesaimtomakethebookappealtoawideraudience and enhance its user-friendly features. • A new chapter (Chap.10) has been added, which presents some basic numerical methods (Euler, Euler midpoint, improved Euler, and fourth-order Runge–Kutta) for approximating the solutions of first-order initial value problems. The exercises in Sects.10.1–10.3 and the first ten exercises in Sect.10.4 are based on the same problems,sothatstudentscancomparetheresultsproducedbythedifferentmeth- ods. • The text of a few sections has been augmented with additional comments, expla- nations, and examples. • AppendixB4hasbeeninserted,toremindusersthedefinitionandbasicproperties of hyperbolic functions. • The number of worked examples has been increased from 232 to 246. • The number of exercises has been increased from 810 to 1010. • All the misprints/omissions detected in the first edition have been corrected. In Chap.10, numerical results are rounded to the fourth decimal place. Also, to avoidcumbersomenotation,andwithoutdangerofambiguity,theapproximateequality symbol has been replaced by the equality sign. I would like to thank Elizabeth Loew, the executive editor for mathematics at Springer, New York, for her guidance during the completion of this project and, in alphabetical order, Kimberly Adams, Peyton Cook, Matteo Dalla Riva, Matthew Don- ahue, Dale Doty, William Hamill, Shirley Pomeranz, and Dragan Skropanic for con- structive comments and suggestions and for their help with proofreading parts of the manuscript. Finally,myspecialgratitudegoestomywife,whoacceptsthatforsomemathemati- cians, writing books is an incurable affliction, whose sufferers are in need of constant support and understanding. Tulsa, OK, USA Christian Constanda January 2017 vii Preface to the First Edition Arguably, one of the principles underpinning classroom success is that the instructor always knows best. Whether this mildly dictatorial premise is correct or not, it seems logical that performance can only improve if the said instructor also pays attention to customerfeedback.Students’opinionsometimescontainsvaluablepointsand,properly canvassed and interpreted, may exercise a positive influence on the quality of a course and the manner of its teaching. When I polled my students about what they wanted from a textbook, their answers clustered around five main issues. The book should be easy to follow without being excessively verbose. A crisp, con- cise, and to-the-point style is much preferred to long-winded explanations that tend to obscure the topic and make the reader lose the thread of the argument. The book should not talk down to the readers. Students feel slighted when they are treatedasiftheyhavenobasicknowledgeofmathematics,andmanyregardthemulti- colored, heavily illustrated texts as better suited for inexperienced high schoolers than for second-year university undergraduates. Thebookshouldkeepthetheorytoaminimum.Lengthyandconvolutedproofsshould be dropped in favor of a wide variety of illustrative examples and practice exercises. Thebookshouldnotembedcomputationaldevicesintheinstructionprocess.Although born in the age of the computer, a majority of students candidly admit that they do not learn much from electronic number crunching. The book should be “slim.” The size and weight of a 500-page volume tend to dis- courage potential readers and bode ill for its selling price. In my view, a book that tries to be “all things to all men” often ends up disappoint- ing its intended audience, who might derive greater profit from a less ambitious but more focused text composed with a twist of pragmatism. The textbooks on differential equations currently on the market, while professionally written and very comprehen- sive,fail,Ibelieve,onatleastoneoftheabovecriteria;bycontrast,thisbookattempts to comply with the entire set. To what extent it has succeeded is for the end user to decide.AllIcansayatthisstageisthatstudentsinmyinstitutionandelsewhere,hav- ing adopted an earlier draft as prescribed text, declared themselves fully satisfied by it and agreed that every one of the goals on the above wish list had been met. The final versionincorporatesseveraladditionsandchangesthatanswersomeoftheircomments and a number of suggestions received from other colleagues involved in the teaching of the subject. Inearliertimes,mathematicalanalysiswastackledfromtheoutsetwithwhatiscalled the ε–δ methodology. Those times are now long gone. Today, with a few exceptions, all ix x Preface to the FirstEdition scienceandengineeringstudents,includingmathematicsmajors,startbygoingthrough calculus I, II, and III, where they learn the mechanics of differentiation and integration but are not shown the proofs of some of the statements in which the formal techniques are rooted, because they have not been exposed yet to the (cid:4)–δ language. Those who wanttoseetheseproofsenrollinadvancedcalculus.Consequently,thenaturalcontinu- ation of the primary calculus sequence for all students is a differential equations course that teaches them solution techniques without the proofs of a number of fundamental theorems. The missing proofs are discussed later in an advanced differential equations sequel (compulsory for mathematics majors and optional for the interested engineering students), where they are developed with the help of advanced calculus concepts. This bookisintendedforusewiththefirst—elementary—differentialequationscourse,taken by mathematics, physics, and engineering students alike. Omittedproofsaside,everybuildingblockofeverymethoddescribedinthistextbook is assembled with total rigor and accuracy. Thebookiswritteninastylethatuseswords(sparingly)asabondingagentbetween consecutive mathematical passages, keeping the author’s presence in the background and allowing the mathematics to be the dominant voice. This should help the readers navigate the material quite comfortably on their own. After the first examples in each section or subsection are solved with full details, the solutions to the rest of them are presented more succinctly: every intermediate stage is explained, but incidental computation (integration by parts or by substitution, finding the roots of polynomial equations, etc.) is entrusted to the students, who have learned the basics of calculus and algebra and should thus be able to perform it routinely. Thecontents,somewhatinexcessofwhatcanbecoveredduringonesemester,include all the classical topics expected to be found in a first course on ordinary differential equations. Numerical methods are off the ingredient list since, in my view, they fall under the jurisdiction of numerical analysis. Besides, students are already acquainted with such approximating procedures from calculus II, where they are introduced to Euler’s method. Graphs are used only occasionally, to offer help with less intuitive concepts (for instance, the stability of an equilibrium solution) and not to present a visual image of the solution of every example. If the students are interested in the latter, they can generate it themselves in the computer lab, where qualified guidance is normally provided. The book formally splits the “pure” and “applied” sides of the subject by placing theinvestigationofselectedmathematicalmodelsinseparatechapters.Boundaryvalue problemsaretoucheduponbriefly(forthebenefitoftheundergraduateswhointendto go on to study partial differential equations), but without reference to Sturm–Liouville analysis. Althoughonlyabout260pageslong,thebookcontains232workedexamplesand810 exercises. There is no duplication among the examples: no two of them are of exactly the same kind, as they are intended to make the user understand how the methods are applied in a variety of circumstances. The exercises aim to cement this knowledge and areallsuitableashomework;indeed,eachandeveryoneofthemispartofmystudents’ assignments. Computer algebra software—specifically, Mathematica(cid:2)R—is employed in the book for only one purpose: to show how to verify quickly that the solutions obtained are correct.Since,inspiteofitsname,thispackagehasnotbeencreatedbymathematicians, it does not always do what a mathematician wants. In many other respects, it is a perfectly good instrument, which, it is hoped, will keep on improving so that when, say, version 54 is released, all existing deficiencies will have been eliminated. I take the view that to learn mathematics properly, one must use pencil and paper and solve Preface to the First Edition xi problems by brain and hand alone. To encourage and facilitate this process, almost all the examples and exercises in the book have been constructed with integers and a few easily managed fractions as coefficients and constant terms. Truth be told, it often seems that the aim of the average student in any course these days is to do just enough to pass it and earn the credits. This book provides such students with everything they need to reach their goal. The gifted ones, who are interested not only in the how but also in the why of mathematical methods and try hard to improve from a routinely achieved 95% on their tests to a full 100%, can use the book as a springboard for progress to more specialized sources (see the list under Further Reading) or for joining an advanced course where the theoretical aspects left out of the basic one are thoroughly investigated and explained. And now, two side issues related to mathematics, though not necessarily to differen- tial equations. Scientists,andespeciallymathematicians,needintheirworkmoresymbolsthanthe Latinalphabethastooffer.Thisforcesthemtoborrowfromotherscripts,amongwhich Greek is the runaway favorite. However, academics—even English speaking ones— cannot agree on a common pronunciation of the Greek letters. My choice is to go to the source, so to speak, and simply follow the way of the Greeks themselves. If anyone else is tempted to try my solution, they can find details in Appendix D. Manyinstructorswouldprobablyagreethatoneofthereasonswhysomestudentsdo notgetthehighgradestheyaspiretoisacocktailofannoyingbadhabitsandincorrect algebra and calculus manipulation “techniques” acquired (along with the misuse of the word “like”) in elementary school. My book Dude, Can You Count? (Copernicus, Springer, 2009) systematically collects the most common of these bloopers and shows howanynumberofabsurditiescanbe“proved”ifsucherrorsareacceptedaslegitimate mathematical handling. Dude is a recommended reading for my classroom attendees, who,Iampleasedtoreport,nowcommitfarfewererrorsintheirwrittenpresentations than they used to. Alas, the cure for the “like” affliction continues to elude me. This book would not have seen the light of day without the special assistance that I received from Elizabeth Loew, my mathematics editor at Springer–New York. She monitoredtheevolutionofthemanuscriptateverystage,offeredadviceandencourage- ment,andwasparticularlyunderstandingoverdeadlines.Iwishtoexpressmygratitude to her for all the help she gave me during the completion of this project. IamalsoindebtedtomycolleaguesPeytonCookandKimberlyAdams,whotrawled the text for errors and misprints and made very useful remarks, to Geoffrey Price for useful discussions, and to Dale Doty, our departmental Mathematica(cid:2)R guru. (Readers interested in finding out more about this software are directed to the website http:// www.wolfram.com/mathematica/.) Finally, I want to acknowledge my students for their interest in working through all the examples and exercises and for flagging up anything that caught their attention as being inaccurate or incomplete. My wife, of course, receives the highest accolade for her inspiring professionalism, patience, and steadfast support. Tulsa, OK, USA Christian Constanda April 2013 Contents 1 Introduction 1 1.1 Calculus Prerequisites............................................. 1 1.2 Differential Equations and Their Solutions........................... 3 1.3 Initial and Boundary Conditions ................................... 6 1.4 Classification of Differential Equations .............................. 9 2 First-Order Equations 15 2.1 Separable Equations .............................................. 15 2.2 Linear Equations ................................................. 20 2.3 Homogeneous Polar Equations ..................................... 24 2.4 Bernoulli Equations............................................... 27 2.5 Riccati Equations ................................................ 29 2.6 Exact Equations ................................................. 31 2.7 Existence and Uniqueness Theorems ................................ 36 2.8 Direction Fields .................................................. 41 3 Mathematical Models with First-Order Equations 43 3.1 Models with Separable Equations................................... 43 3.2 Models with Linear Equations...................................... 46 3.3 Autonomous Equations............................................ 51 4 Linear Second-Order Equations 63 4.1 Mathematical Models with Second-Order Equations .................. 63 4.2 Algebra Prerequisites ............................................. 64 4.3 Homogeneous Equations........................................... 68 4.3.1 Initial Value Problems ...................................... 69 4.3.2 Boundary Value Problems ................................... 71 4.4 Homogeneous Equations with Constant Coefficients................... 74 4.4.1 Real and Distinct Characteristic Roots........................ 74 4.4.2 Repeated Characteristic Roots ............................... 77 4.4.3 Complex Conjugate Characteristic Roots ...................... 80 4.5 Nonhomogeneous Equations ....................................... 83 4.5.1 Method of Undetermined Coefficients: Simple Cases ............ 83 xiii xiv Contents 4.5.2 Method of Undetermined Coefficients: General Case ............ 90 4.5.3 Method of Variation of Parameters ........................... 95 4.6 Cauchy–Euler Equations .......................................... 98 4.7 Nonlinear Equations .............................................. 102 5 Mathematical Models with Second-Order Equations 105 5.1 Free Mechanical Oscillations ....................................... 105 5.1.1 Undamped Free Oscillations ................................. 105 5.1.2 Damped Free Oscillations ................................... 108 5.2 Forced Mechanical Oscillations ..................................... 111 5.2.1 Undamped Forced Oscillations ............................... 111 5.2.2 Damped Forced Oscillations ................................. 113 5.3 Electrical Vibrations .............................................. 115 6 Higher-Order Linear Equations 119 6.1 Modeling with Higher-Order Equations.............................. 119 6.2 Algebra Prerequisites ............................................. 119 6.2.1 Matrices and Determinants of Higher Order ................... 120 6.2.2 Systems of Linear Algebraic Equations ........................ 121 6.2.3 Linear Independence and the Wronskian ...................... 125 6.3 Homogeneous Differential Equations ................................ 128 6.4 Nonhomogeneous Equations ....................................... 132 6.4.1 Method of Undetermined Coefficients ......................... 132 6.4.2 Method of Variation of Parameters ........................... 136 7 Systems of Differential Equations 139 7.1 Modeling with Systems of Equations ................................ 139 7.2 Algebra Prerequisites ............................................. 141 7.2.1 Operations with Matrices.................................... 141 7.2.2 Linear Independence and the Wronskian ...................... 147 7.2.3 Eigenvalues and Eigenvectors ................................ 149 7.3 Systems of First-Order Differential Equations ........................ 152 7.4 Homogeneous Linear Systems with Constant Coefficients .............. 155 7.4.1 Real and Distinct Eigenvalues................................ 157 7.4.2 Complex Conjugate Eigenvalues.............................. 162 7.4.3 Repeated Eigenvalues ....................................... 167 7.5 Other Features of Homogeneous Linear Systems ...................... 174 7.6 Nonhomogeneous Linear Systems................................... 179 8 The Laplace Transformation 189 8.1 Definition and Basic Properties .................................... 189 8.2 Further Properties................................................ 195 8.3 Solution of IVPs for Single Equations ............................... 200 8.3.1 Continuous Forcing Terms................................... 200 8.3.2 Piecewise Continuous Forcing Terms.......................... 205 8.3.3 Forcing Terms with the Dirac Delta........................... 208 8.3.4 Equations with Variable Coefficients .......................... 212 8.4 Solution of IVPs for Systems....................................... 215