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INTRODUCTORY DIFFERENTIAL EQUATIONS INTRODUCTORY DIFFERENTIAL EQUATIONS FIFTH EDITION Martha L. Abell GeorgiaSouthernUniversity,Statesboro,GA,USA James P. Braselton GeorgiaSouthernUniversity,Statesboro,GA,USA AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1800,SanDiego,CA92101-4495,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2018ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,including photocopying,recording,oranyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher.Details onhowtoseekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizations suchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-814948-5 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:KateyBirtcher AcquisitionEditor:KateyBirtcher EditorialProjectManager:SusanIkeda ProductionProjectManager:PaulPrasadChandramohan Designer:MarkRogers TypesetbyVTeX Preface Introductory Differential Equations began as computer algebra system so that they could be a text originally called Introductory Differential exposed to the technology. However, we soon EquationswithBoundaryValueProblems. realized that we were missing the great op- When we were done with the revision, we portunity of allowing students to discover as- no longer saw Introductory Differential Equations pects of the subject matter on their own. We with Boundary Value Problems but rather a new revised our materials to include experimental text that we have titled Introductory Differential problems and thought-provoking questions in Equations. which students are asked to make conjectures The first edition of this text, Modern Differen- and investigate supporting evidence. We also tialEquationswas“modern”becauseitwasone developedapplicationprojectscalledDifferential ofthefirsttextsthatrequiredaccesstoagraph- Equations at Work, not only to emphasize tech- ing calculator, computer algebra system, or nu- nology,butalsotoimprovetheproblem-solving mericalsoftwarepackage. and communication skills of our students. To Computeralgebrasystemsandsophisticated preserve the “wow” aspects of technology, we graphing calculators have changed the ways in continue to use it to observe solutions in class- which we learn and teach ordinary differential roomdemonstrationsthroughsuchthingsasan- equations. Instead of focusing students’ atten- imating the motion of springs and pendulums. tion only on a sequence of solution methods, These demonstrations not only grab the atten- wewantthemtousetheirmindstounderstand tionofstudents,butalsohelpthemtomakethe whatsolutionsmeanandhowdifferentialequa- connection between a formula and what it rep- tionscanbeusedtoanswerpertinentquestions. resents. Nowtheiruseisexpectedinastandardcourse, This text is designed to serve as a text for so the term “modern” no longer applies to the beginningcoursesinordinarydifferentialequa- text. tions.Usually,introductoryordinarydifferential Interestingly, this metamorphosis in the equations courses are taken by students who teaching of differential equations described oc- have successfully completed a first-year calcu- curredrelativelyquicklyandcoincidedwithour luscourse,abasiclinearalgebracourse,andthis professional careers at Georgia Southern Uni- textiswrittenatalevelreadableforthem. versity. Our interest in the use of technology in the mathematics classroom began in 1990 when we started to use computer laboratories TECHNOLOGY and demonstrations in our calculus, differen- tialequations,andappliedmathematicscourses. Some advantages of incorporating technol- Overthepastyearswehavelearnedsomeways ogy into mathematics courses include enhanc- of how to and how not to use technology in ing the ability to solve a variety of problems, themathematicscurriculum.Intheearlystages, helpingstudentsworkexamples,supportingin- we simply wanted to show students how they terestingapplications,exploitingandimproving could solve more difficult problems by using a geometric intuition, encouraging mathematical vii viii PREFACE experiments, and helping students understand data. In particular, obtaining closed form solu- approximation.Inaddition,technologyisimple- tionsisnotnecessarily“easy”(oralwayspossi- mentedthroughoutthistexttopromotethefol- ble).Theseapplications,evenifnotformallydis- lowinggoalsinthelearningofdifferentialequa- cussed in class, show students that differential tions: equations is an exciting and interesting subject withextensiveapplicationsinmanyfields. 1. Solvingproblems:Usingdifferentmethodsto solveproblemsandgeneralizesolutions; 2. Reasoning: Exploiting computer graphics to STYLE develop spatial reasoning through visualization; To keep the text as flexible as possible, ad- 3. Analyzing:Findingthemostreasonablesolu- dressingtheneedsofbothaudienceswithdiffer- tiontorealproblemsorobservingchangesin ent mathematical backgrounds and instructors thesolutionunderchangingconditions; with varying preferences, Introductory Differen- 4. Communicating mathematics: Developing tial Equations is written in an easy-to-read, yet written, verbal, and visual skills to commu- mathematicallyprecise,style.Itcontainsalltop- nicatemathematicalideas;and ics typically included in a standard first course 5. Synthesizing:Makinginferencesandgeneral- in ordinary differential equations. Definitions, izations,evaluatingoutcomes,classifyingob- theorems, and proofs are concise but worded jects,andcontrollingvariables. precisely for mathematical accuracy Generally, Students who develop these skills will suc- theorems are proved if the proof is instructive ceed not only in differential equations, but also orhas“teachingvalue.”Ofcourse,discussionof insubsequentcoursesandintheworkforce. such proofs is optional in the typical classroom for which this text is written. In other cases, The icon is used throughout the text to proofs of theorems are developed in the exer- indicate those examples in which technology is cises or omitted. Theorems and definitions are used in a nontrivial way to develop or visual- boxed for easy reference; key terms are high- izethesolutionortoindicatethesectionsofthe lighted in boldface. Figures are used frequently text, such as those discussing numerical meth- to clarify material with a graphical interpreta- ods,inwhichtheuseofappropriatetechnology tion. isessentialorinteresting. FEATURES APPLICATIONS Introductory Differential Equations, Fifth edi- Applicationsinthistextaretakenfromava- tion, is an extensive revision of the Fourth edi- rietyoffields,especiallybiology,physics,chem- tion of Introductory Differential Equations. This istry, engineering, and economics, and they are editionincludesseveralofthefollowing: documented by references. These applications can be found in many of the examples and ex- • Because the mathematician’s who developed ercises, in separate sections and chapters of the the mathematics discussed in this text were text,andintheDifferentialEquationsatWorksub- (orare)stillinterestingintheirownright,we sections at the end of each chapter. Many of have tried to include an image and interest- theseapplicationsarewell-suitedtoexploration ingtidbitabouttheirliveswheneverpossible with technology because they incorporate real hoping to help some students become more ix PEDAGOGICALFEATURES interested in the course. When credit is not Technology given for a photo, it is because we have rea- Many students entering their first differen- sonablereasontobelievethattheimageisin tialcoursehavehadsubstantialexperiencewith the public domain. If a copyright applies to various sophisticated calculators and computer animageandappropriatecredithasnotbeen algebrasystems. given, please alert us so that we can correct Our first ordinary differential equations thesituationpromptly. course attempts to encourage students to use • All graphics have been redone. In each case, technology intelligently. We have italicized the theintentofthegraphichasbeenquestioned. words use technology intelligently because they Insomecases,graphicshavebeeneliminated; take on different meaning to different instruc- in other cases, they have been redone to em- tors because they depend on the instructor’s phasize their purpose or new graphics have philosophy, institution, and students. Students beenintroducedtothetext. also interpret the phrase differently depending • Wehaverevisedtheexercisesetconsiderably. upon their instructor and exposure to technol- The total number of exercises remains about ogy. thesameasinpreviouseditionsbutwehave In any case, many instructors have limited deleted about 100 “outdated” exercises and resources and would prefer that our students replacedthemwithnewones.Wecontinueto have a good grasp of the fundamentals rather believethatastudentshouldbeabletosolve than be “wowed!” by nonsense. We have tried abasicnontrivialproblembyhand. to use technology intelligently. We believe that it should not be obtrusive so you should not notice when we do. When required in an ex- PEDAGOGICAL FEATURES ample or exercise, it should be obvious to an instructorandrelativelyeasytoconvinceastu- Examples dentthattherearetwowaystosolveaproblem: the easy way and the hard way. We choose the Throughoutthetext,numerousexamplesare hard way when there is instructional value to given, with thorough explanations and a sub- stantialamountofdetail.Solutionstomoredif- the approach. The icon is intended to alert ficultexamplesareconstructedwiththehelpof students that technology is intelligently (and graphingcalculatorsoracomputeralgebrasys- wisely) used to assist in solving the problem. temandareindicatedbyanicon. Typically,thetechnologywehaveusedisacom- puteralgebrasystem,likeMathematicaorMaple. “Think about it!” Technologyisusedthroughoutthetexttoex- plore many of the applications and more diffi- Many examples are followed by a question cultexamples,especiallythosemarkedwith indicatedbya icon.Generally,basicknowl- and the problems in the subsections Differential edge about the behavior of functions is suffi- EquationsatWork. cient to answer the question. Many of these Answers to most odd exercises are included questionsencouragestudentstousetechnology. at the end of the text. More complete answers, Others, focus on the graph of a solution. Thus, solutions, partial solutions, or hints to selected “Thinkaboutit!”questionshelpstudentsdeter- exercises are available separately to students minewhentousetechnologyandmakethistext and instructors. Differential Equations at Work moreinteractive. subsections describe detailed economics, biol- x PREFACE ogy, physics, chemistry, and engineering prob- basic techniques. Most sections also contain in- lemsdocumentedbyreferences.Theseproblems terestingmathematicalandappliedproblemsto include real data when available and require show that mathematics and its applications are students to provide answers based on different both interesting and relevant. Instructors will conditions. Students must analyze the problem findthattheycanassignalargenumberofprob- andmakedecisionsaboutthebestwaytosolve lems,ifdesiredyetstillhaveplentyforreviewin it, including the appropriate use of technology additiontothosefoundinthereviewsectionat EachDifferentialEquationsatWorkprojectcanbe the end of each chapter. Answers to most odd- assignedasaprojectrequiringawrittenreport, numbered exercises are included at the end of forgroupwork,orfordiscussioninclass. the text; detailed solutions to selected exercises Differential Equations at Work also illustrate areincludedintheStudentResourceManual. how differential equations are used in the real world. Students are often reluctant to believe Chapter Summary and Review Exercises that the subject matter in calculus, linear algebra, and differential equations classes relates to Each chapter ends with a chapter summary subsequent courses and to their careers. Each highlightingimportantconcepts,keytermsand Differential Equations at Work subsection illus- formulas, and theorems. The Review Exercises trateshowthematerialdiscussedinthecourseis followingthechaptersummaryofeachchapter usedinreallife.WekeepeachDifferentialEqua- offerstudentsextrapracticeonthetopicsinthat tions at Work subsection short because nearly chapter.Theseexercisesarearrangedbysection allinstructorshaveenoughtroublecoveringthe sothatstudentshavingdifficultycanturntothe content expected of them. On the other hand, appropriatematerialforreview. when a student asks the question “When am I going to use this?” or “How am I going to use this?” these short subsections can give the in- Figures structorideasastohowtohandlethequestion. This text provides an abundance of figures TheproblemsinDifferentialEquationsatWork andgraphs,especiallyforsolutionstoexamples. are not connected to a specific section of the Inaddition,studentsareencouragedtodevelop text; they require students to draw different spatial visualization and reasoning skills, to in- mathematical skills and concepts together to terpretgraphs,andtodiscoverandexplorecon- solve a problem. BecauseeachDifferential Equa- tions at Work is cumulative in nature, students cepts from a graphical point of view. To ensure must combine mathematical concepts, tech- accuracy,thefiguresandgraphshavebeencom- niques,andexperiencesfrompreviouschapters pletely computer-generated. We hope you like andmathcourses. theimprovementinthegraphicsinfifthedition ofthetext. Exercises Historical Material Numerous exercises, ranging in level from easy to difficult, are included in each section of Nearly every topic is motivated by either an thetext.Inparticular,theexercisesetsfortopics applicationoranappropriatehistoricalnote.We thatstudentsfindmostdifficultarerichandvar- have also included images of paintings, draw- ied.Forthefourtheditionofthetext,theabun- ings, or photographs of the many famous sci- dant “routine” exercises have been completely entistsanddescriptionsofthemathematicsthey revisedandtrytoencouragestudentstomaster discovered. xi CONTENT CONTENT duced,whichincludesadiscussionofsev- eral special equations and the properties The highlights of each chapter are described of their solutions/equations important in brieflybelow. many areas of applied mathematics and Chapter1 After introducing preliminary def- physics. initions, we discuss direction fields not Chapter5 Several applications of higher or- only for first-order differential equations, der equations are presented. The distinc- but also for systems of equations. In this tive presentation illustrates the motion presentation, we establish a basic under- of spring-mass systems and pendulums standingofsolutionsandtheirgraphs.We graphically to help students understand give an overview of some of the applica- what solutions represent and to make the tions covered later in the text to point out applicationsmoremeaningfultothem. theusefulnessofthetopicandsomeofthe Chapter6 The study of systems of differential reasons we have for studying differential equations is perhaps the most exciting of equationsintheexercises. allthetopicscoveredinthetext.Although Chapter2 In addition to discussing the stan- we direct most of our attention to solv- dard techniques for solving several types ing systems of linear first-order equations of first-order differential equations (sepa- with constant coefficients, technology al- rable equations, homogeneous equations, lows us to investigate systems of nonlin- exactequations,andlinearequations),we ear equations and observe phase planes. introduceseveralnumericalmethods(Eu- Wealsoshowhowtouseeigenvaluesand ler’s Method, Improved Euler’s Method, eigenvectorstounderstandthegeneralbe- Runge-Kutta Method) and discuss the ex- havior of systems of linear and nonlinear istenceanduniquenessofsolutionstofirst equations. We have added a section on order initial-value problems. Throughout phaseportraitsinthisedition. the chapter, we encourage students to Chapter7 Several applications discussed ear- buildanintuitiveapproachtothesolution lier in the text are extended to more than processbymatchingagraphtoasolution one dimension and solved using systems withoutactuallysolvingtheequation. ofdifferentialequations,inanefforttore- Chapter3 Not only do we cover most stan- inforce the understanding of these impor- dard applications of first order equations tant problems.Numerous applications in- inChapter3(orthogonaltrajectories,pop- volving nonlinear systems are discussed ulation growth and decay, Newton’s law aswell. of cooling, free-falling bodies), but we Chapter8 Laplacetransformsareimportantin also present many that are not (due to many areasofengineering andexhibit in- their computational difficulty) in Differen- triguing mathematical properties as well. tialEquationsatWork. Throughout the chapter, we point out the Chapter4 This chapter emphasizes the meth- importance of initial conditions and forc- odsforsolvinghomogeneousandnonho- ingfunctionsoninitial-valueproblems. mogeneoushigherorderdifferentialequa- For a one semester course introducing or- tions.ItalsostressesthePrincipleofSuper- dinary differential equations, many instructors position and the differences between the willchoosetocovertopicsfromChapters1to7 properties of solutions to linear and non- orfromChapters1to6andChapter8.Foratwo linearequations.AfterdiscussingCauchy- semestercourse,theinstructorwilleasilybeable Euler equations, series methods are intro- to cover the remaining chapters of the text. In