Introductory differential equations with boundary value problems

520 Pages·2015·35.77 MB·English

by  Martha L Abell

INTRODUCTORY
DIFFERENTIAL
EQUATIONS
INTRODUCTORY
DIFFERENTIAL
EQUATIONS
FOURTH EDITION
Martha L. Abell
GeorgiaSouthernUniversity,Statesboro,Georgia,USA
James P. Braselton
GeorgiaSouthernUniversity,Statesboro,Georgia,USA
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ISBN:978-0-12-417219-7
LibraryofCongressCataloging-in-PublicationData
Abell,MarthaL.,1962-
Introductorydifferentialequations/MarthaL.Abell,JamesP.Braselton.–Fourthedition.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-12-417219-7(alk.paper)
1.Differentialequations–Textbooks.I.Braselton,JamesP.,1965-II.Title.III.Title:Differentialequations
withboundaryvalueproblems.
QA379.A242014
515’.35–dc23
2014006809
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Preface
Introductory Differential Equations began as we simply wanted to show students how they
the fourth edition of a text originally called could solve more difficult problems by using
Introductory Differential Equations with Boundary a computer algebra system so that they could
ValueProblems. be exposed to the technology. However, we
When we were done with the revision, we soon realized that we were missing the great
no longer saw Introductory Differential Equations opportunity of allowing students to discover
with Boundary Value Problems but rather a new aspects of the subject matter on their own. We
text that we have titled Introductory Differential revised our materials to include experimental
Equations. problems and thought-provoking questions in
Thefirsteditionofthistext,ModernDifferential which students are asked to make conjectures
Equations was “modern” because it was one of and investigate supporting evidence. We also
the first texts that nearly required access to a developedapplicationprojectscalledDifferential
graphing calculator, computer algebra system, Equations at Work, not only to emphasize
ornumericalsoftwarepackage. technology, but also to improve the problem-
Computer algebrasystemsandsophisticated solving and communication skills of our
graphing calculators have changed the ways in students. To preserve the “wow” aspects of
which we learn and teach ordinary differential technology, we continue to use it to observe
equations. Instead of focusing students’ atten- solutions in classroom demonstrations through
tion only on a sequence of solution methods, such things as animating the motion of springs
wewantthemtousetheirmindstounderstand andpendulums.Thesedemonstrationsnotonly
whatsolutionsmeanandhowdifferentialequa- grabtheattentionofstudents,butalsohelpthem
tionscanbeusedtoanswerpertinentquestions. to make the connection between a formula and
Nowtheir use isexpected inastandard course, whatitrepresents.
so the term “modern” no longer applies to the This text is designed to serve as a text for
text. beginningcoursesinordinarydifferentialequa-
Interestingly, this metamorphosis in the tions.Usually,introductoryordinarydifferential
teaching of differential equations described equations courses are taken by students who
occurred relatively quickly and coincided with havesuccessfullycompletedafirst-yearcalculus
our professional careers at Georgia Southern course, a basic linear algebra course, and this
University.Ourinterestintheuseoftechnology text is written at a level readable for them.
in the mathematics classroom began in 1990 Previous versions of the texts have included
when we started to use computer laboratories two chapters on boundary value problems and
anddemonstrations inourcalculus, differential partial differential equations. Often, the first
equations, and applied mathematics courses. ordinary differential equations course and the
Overthepastyearswehavelearnedsomeways first partial differential equations course are
of how to and how not to use technology in completely separated. We believe that there
themathematicscurriculum.Intheearlystages, are many more comprehensive texts for a first
vii
viii
PREFACE
course in partial differential equations than thesolutionortoindicatethesectionsofthetext,
the previous two chapters on boundary value such asthose discussing numericalmethods, in
problems and partialdifferential equations that which the use of appropriate technology is es-
were included in this text. Of course, if any sentialorinteresting.
instructororstudentusingthistextwantsaccess
to those two chapters for their classes we are
APPLICATIONS
morethanhappytosupplythemtoyou.
Applications in this text are taken from a
TECHNOLOGY variety of fields, especially biology, physics,
chemistry,engineering,andeconomics,andthey
are documented by references. These applica-
The advantages of incorporating technology
tions can be found in many of the examples
intomathematicscoursesarewell-known.Some
and exercises, in separate sections and chapters
of them include enhancing the ability to solve
of the text, and in the Diffirential Equations at
a variety of problems; helping students work
Work subsections at the end of each chapter.
examples; supporting varied, realistic, and illu-
Many of these applications are well-suited
minatingapplications;exploitingandimproving
to exploration with technology because they
geometric intuition; encouraging mathematical
incorporate real data. In particular, obtaining
experiments;andteachingapproximation.Inad-
closed form solutions is not necessarily “easy”
dition, technology is implemented throughout
(oralwayspossible). Theseapplications, evenif
this text to promote the following goals in the
not formally discussed in class, show students
learningofdifferentialequations:
that differential equations is an exciting and
1. Solvingproblems:Usingdifferentmethods interesting subject with extensive applications
tosolveproblemsandgeneralizesolutions; inmanyfields.
2. Reasoning:Exploitingcomputergraphicsto
developspatialreasoningthrough
STYLE
visualization;
3. Analyzing:Findingthemostreasonable
solutiontorealproblemsorobserving To keep the text as flexible as possible,
changesinthesolutionunderchanging addressing the needs of both audiences with
conditions; different mathematical backgrounds and in-
4. Communicatingmathematics:Developing structors with varying preferences, Introductory
written,verbal,andvisualskillsto Differential Equations is written in an easy-
communicatemathematicalideas;and to-read, yet mathematically precise, style. It
5. Synthesizing:Makinginferencesand contains all topics typically included in a
generalizations,evaluatingoutcomes, standard first course in ordinary differential
classifyingobjects,andcontrollingvariables. equations.Definitions,theorems,andproofsare
concise but worded precisely for mathematical
Students who develop these skills will suc-
accuracy Generally, theorems are proved if the
ceed not only in differential equations, but also
proof is instructive or has “teaching value.” Of
insubsequentcoursesandintheworkforce.
course, discussion of such proofs is optional
in the typical classroom for which this text is
The iconisusedthroughout thetextto written. In other cases, proofs of theorems are
indicate those examples in which technology is developedintheexercisesoromitted.Theorems
usedinanontrivialwaytodeveloporvisualize and definitions are boxed for easy reference;
ix
PREFACE
key terms are highlighted in boldface. Figures PEDAGOGICAL FEATURES
are used frequently to clarify material with a
graphicalinterpretation. Examples
Throughout thetext,numerousexamplesare
given, with thorough explanations and a sub-
FEATURES
stantialamountof detail.Solutions tomore dif-
ficult examples are constructed with the help
Introductory Differential Equations is an exten- of graphing calculators or a computer algebra
sive revision of the third edition of Introductory systemandareindicatedbyanicon.
Differential Equations with Boundary Value Prob-
lems.Particularfeaturesinclude:
“Think About It!”
• Thetext’swebsiteincludebackground
material,proofsofsometheorems,solutions
Many examples are followed by a question
toselectedexercises,additionalexercises,
indicated by a icon. Generally, basic
visualizationsofcertaintopics(movies)and
podcaststhatstudentscandownloadtotheir
knowledge about the behavior of functions
video-capableiPodorothercompatiblemv4
is sufficient to answer the question. Many
players.
of these questions encourage students to use
• Becausethemathematician’swhodeveloped
technology. Others, focus on the graph of a
themathematicsdiscussedinthistextwere
solution. Thus,“Thinkaboutit!”questionshelp
(orare)stillinterestingintheirownright,we
studentsdeterminewhentousetechnologyand
havetriedtoincludeanimageand
makethistextmoreinteractive.
interestingtidbitabouttheirliveswhenever
possiblehopingtohelpsomestudents
becomemoreinterestedinthecourse.When
Technology
creditisnotgivenforaphoto,itisbecause
wehavereasonablereasontobelievethatthe Many students entering their first differen-
imageisinthepublicdomain.Ifacopyright tialcoursehavehadsubstantialexperiencewith
appliestoanimageandappropriatecredit various sophisticated calculators and computer
hasnotbeengiven,pleasealertussothatwe algebrasystems.
cancorrectthesituationpromptly Our first ordinary differential equations
• Allgraphicshavebeenredone.Ineachcase, course attempts to encourage students to use
theintentofthegraphichasbeen technology intelligently. We have italicized the
questioned.Insomecases,graphicshave words use technology intelligently because they
beeneliminated;inothercases,theyhave takeondifferentmeaningtodifferentinstructors
beenredonetoemphasizetheirpurpose. becausetheydepend ontheinstructor’s philos-
• Wehaverevisedtheexerciseset ophy, institution, and students. Students also
considerably.Thetotalnumberofexercises interpretthephrasedifferentlydependingupon
remainsaboutthesameasinprevious theirinstructorandexposuretotechnology.
editionsbutwehavedeletedabout250 Inanycase,manyofushavelimitedresources
“outdated”exercisesandreplacedthemwith and would prefer that our students have a
newones.Wecontinuetobelievethata good grasp of the fundamentals rather than be
studentshouldbeabletosolveabasic “wowed!” by nonsense. We have tried to use
nontrivialproblembyhand. technology intelligently here. We believe that it
x
PREFACE
shouldnotbeobtrusivesoyoushouldnotnotice course is used in real life. We keep each
when we do. When required in an example or Differential Equations at Work subsection short
exercise, it should be obvious to an instructor because nearly all instructors have enough
and relatively easy to convince a student that trouble covering the content expected of them.
there aretwowaystosolve aproblem:the easy On the other hand, when a student asks the
way and the hard way. We choose the hard question “When am I going to use this?” or
way when there is instructional value to the “How am I going to use this?” these short
subsections can give the instructor ideas as to
approach. The icon is intended to alert
howtohandlethequestion.
students that technology is intelligently (and TheproblemsinDifferentialEquationsatWork
wisely) used to assist in solving the problem. are not connected to a specific section of the
Typically, the technology we have used is a text; they require students to draw different
computer algebra system, like Mathematica or mathematical skills and concepts together
Maple. to solve a problem. Because each Differential
Technologyisusedthroughoutthetexttoex- Equations at Work is cumulative in nature,
ploremanyoftheapplicationsandmoredifficult students must combine mathematical concepts,
techniques, and experiences from previous
chaptersandmathcourses.
examples, especially those marked with
and the problems in the subsections Differential
EquationsatWork. Exercises
Answers to most odd exercises are included
Numerous exercises, ranging in level from
at the end of the text. More complete answers,
easy to difficult, are included in each section of
solutions, partial solutions, or hints to selected
thetext.Inparticular,theexercisesetsfortopics
exercisesareavailableseparatelytostudentsand
thatstudentsfindmostdifficultarerichandvar-
instructors. Differential Equations at Work sub-
ied. For the fourthedition of the text, the abun-
sections describe detailed economics, biology,
dant “routine” exercises have been completely
physics, chemistry, and engineering problems
revised and try to encourage students to mas-
documented by references. These problems in-
ter basic techniques. Most sections also contain
clude real data when available and require stu-
interesting mathematicaland applied problems
dentstoprovideanswersbasedondifferentcon-
to show that mathematics and its applications
ditions.Studentsmustanalyzetheproblemand
arebothinterestingandrelevant.Instructorswill
makedecisionsaboutthebestwaytosolveit,in-
findthattheycanassignalargenumberofprob-
cluding the appropriate use of technology Each
lems,ifdesiredyetstillhaveplentyforreviewin
Differential Equations at Work project can be as-
addition tothose found inthe review section at
signedasaprojectrequiringawrittenreport,for
the end of each chapter. Answers to most odd-
groupwork,orfordiscussioninclass.
numbered exercises are included at the end of
Differential Equations at Work also illustrate
the text; detailed solutions to selected exercises
how differential equations are used in the
areincludedintheStudentResourceManual.
real world. Students are often reluctant to
believethatthesubjectmatterincalculus,linear
algebra,anddifferentialequationsclassesrelates Chapter Summary and Review Exercises
to subsequent courses and to their careers.
Each Differential Equations at Work subsection Each chapter ends with a chapter summary
illustrates how the material discussed in the highlighting importantconcepts,keytermsand
xi
PREFACE
formulas, and theorems. The Review Exercises thereasonswehaveforstudyingdifferential
following the chapter summaryof eachchapter equationsintheexercises.
offerstudentsextrapracticeonthetopicsinthat Chapter2Inadditiontodiscussingthe
chapter.Theseexercisesarearrangedbysection standardtechniquesforsolvingseveral
sothatstudentshavingdifficultycanturntothe typesoffirst-orderdifferentialequations
appropriatematerialforreview. (separableequations,homogeneous
equations,exactequations,andlinear
equations),weintroduceseveralnumerical
Figures
methods(Euler’sMethod,ImprovedEuler’s
This text provides an abundance of figures Method,Runge-KuttaMethod)anddiscuss
andgraphs,especiallyforsolutionstoexamples. theexistenceanduniquenessofsolutionsto
Inaddition,studentsareencouragedtodevelop firstorderinitial-valueproblems.
spatial visualization and reasoning skills, to in- Throughoutthechapter,weencourage
terpretgraphs,andtodiscoverandexplorecon- studentstobuildanintuitiveapproachtothe
cepts from a graphical point of view. To ensure solutionprocessbymatchingagraphtoa
accuracy,thefiguresandgraphshavebeencom- solutionwithoutactuallysolvingthe
pletely computer-generated. Nearly all figures equation.
forthefourtheditionhavebeenrevised.Wehope Chapter3Notonlydowecovermost
youliketheimprovementinthegraphics. standardapplicationsoffirstorderequations
inChapter3(orthogonaltrajectories,
populationgrowthanddecay,Newton’slaw
Historical Material
ofcooling,free-fallingbodies),butwealso
presentmanythatarenot(duetotheir
Nearly every topic is motivated by either an
computationaldifficulty)inDifferential
applicationoranappropriatehistoricalnote.We
EquationsatWork.
have also included images of paintings, draw-
Chapter4Thischapteremphasizesthe
ings, or photographs of the many famous sci-
methodsforsolvinghomogeneousand
entistsanddescriptionsofthemathematicsthey
nonhomogeneoushigherorderdifferential
discovered.
equations.ItalsostressesthePrincipleof
Superpositionandthedifferencesbetween
thepropertiesofsolutionstolinearand
CONTENT
nonlinearequations.Afterdiscussing
Cauchy-Eulerequations,seriesmethodsare
The highlights of each chapter are described
introduced,whichincludesadiscussionof
brieflybelow.
severalspecialequationsandtheproperties
Chapter1Afterintroducingpreliminary oftheirsolutions/equations importantin
definitions,wediscussdirectionfieldsnot manyareasofappliedmathematicsand
onlyforfirst-orderdifferentialequations,but physics.
alsoforsystemsofequations.Inthis Chapter5Severalapplicationsofhigher
presentation,weestablishabasic orderequationsarepresented.The
understandingofsolutionsandtheirgraphs. distinctivepresentationillustratesthe
Wegiveanoverviewofsomeofthe motionofspring-masssystemsand
applicationscoveredlaterinthetexttopoint pendulumsgraphicallytohelpstudents
outtheusefulnessofthetopicandsomeof understandwhatsolutionsrepresentandto
xii
PREFACE
maketheapplicationsmoremeaningfulto text. In our introductory ordinary differential
them. equations course, we typically cover most of
Chapter6Thestudyofsystemsof Chapters 1, 2, 4 and 6, and instructors choose a
differentialequationsisperhapsthemost varietyofapplicationsfromChapters3,5and7.
excitingofallthetopicscoveredinthetext.
• Foraonesemestercoursetargetedtothe
Althoughwedirectmostofourattentionto
computationalneedsofmostengineering
solvingsystemsoflinearfirst-order
majors,covermosttopicsfromChapters1,2,
equationswithconstantcoefficients,
4,6and8.
technologyallowsustoinvestigatesystems
• Foracasualonesemestercoursedirected
ofnonlinearequationsandobservephase
towardsmathandmatheducationmajors,
planes.Wealsoshowhowtouseeigenvalues
covermostmathematicaltopicsin
andeigenvectorstounderstandthegeneral
Chapters1,2,4and6andselected
behaviorofsystemsoflinearandnonlinear
applicationsfromChapters3,5and7.
equations.Wehaveaddedasectiononphase
portraitsinthisedition. Finally, we thank those close to us, especially
Chapter7Severalapplicationsdiscussed Imogene Abell, Lori Braselton, Ada Braselton,
earlierinthetextareextendedtomorethan and Mattie Braselton for enduring with us the
onedimensionandsolvedusingsystemsof pressuresofaprojectlikethisandforgraciously
differentialequations,inanefforttoreinforce accepting our demanding work schedules. We
theunderstandingoftheseimportant certainly could not have completed this task
problems.Numerousapplicationsinvolving withouttheircareandunderstanding.
nonlinearsystemsarediscussedaswell.
Chapter8Laplacetransformsareimportant MarthaAbell
inmanyareasofengineeringandexhibit (E-mail:[email protected])
JamesBraselton
intriguingmathematicalpropertiesaswell.
Throughoutthechapter,wepointoutthe (E-mail:jbraselton@
importanceofinitialconditionsandforcing georgiasouthern.edu)
functionsoninitial-valueproblems.
DepartmentofMathematicalSciences
For a one semester course introducing or- P.O.Box8093
dinary differential equations, many instructors GeorgiaSouthernUniversity
willchoosetocovertopicsfromChapters1and7 Statesboro,Georgia
or from Chapters 1 and 6 and Chapter 8. For a 30460
two semester course, the instructor will easily
be able to cover the remaining chapters of the July,2014
C H A P T E R
1
Introduction to Differential Equations
O U T L I N E
1.1 IntroductiontoDifferential Exercises1.2 19
Equations:Vocabulary 3
Chapter1Summary:EssentialConcepts
Exercises1.1 10 andFormulas 22
1.2 AGraphicalApproachtoSolutions: Chapter1ReviewExercises 23
SlopeFieldsandDirectionFields 15
The purpose of Introductory Differential physicalsituationsaremodeledwithdifferential
Equations is twofold. First, we introduce and equations. For example, in Chapter 5, we see
discuss the topics covered in an undergraduate that the motion of a pendulum canbe modeled
courseinordinarydifferentialequations(ODEs). by a differential equation. When we solve the
Second, we indicate how certain technologies problem of the motion of a pendulum, we use
suchascomputeralgebrasystemsandgraphing technology to watch the pendulum move. The
calculators are used to enhance the study of same is true for the motion of a mass attached
differential equations, not only by eliminating to the end of a spring, as well as many other
someofthecomputationaldifficultiesthatarise problems. In having this ability to use technol-
in the study of differential equations but also ogy,thestudyofdifferentialequationsbecomes
by overcoming some of the visual limitations muchmoremeaningfulaswellasinteresting.
associated with the solutions of differential Although this chapter is short in length, the
equations. The advantages of using technology vocabulary introduced here is used throughout
such as graphing calculators and computer the text. To a large extent, this chapter may be
algebra systems in the study of differential read quickly, but it is important to remember
equations are numerous, but perhaps the most that subsequent chapters will take advantage
useful is that of being able to produce the of the terminology and techniques discussed
graphicsassociatedwithsolutionsofdifferential here. Any formal introduction to differential
equations. This is particularly beneficial in equations should begin with German scientist
the discussion of applications because many Gottfried Wilhelm Leibniz (1646-1716) and
IntroductoryDifferentialEquations 1 Copyright©2014ElsevierInc.Allrightsreserved.
http://dx.doi.org/10.1016/B978-0-12-417219-7.00001-6