ELEMENTARY DIFFERENTIAL EQUATIONS

663 Pages·2013·7.84 MB·English

ELEMENTARY DIFFERENTIAL
EQUATIONS
William F. Trench
Andrew G.CowlesDistinguishedProfessor Emeritus
DepartmentofMathematics
TrinityUniversity
SanAntonio,Texas, USA
[email protected]
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FREE DOWNLOAD:STUDENTSOLUTIONSMANUAL
FreeEdition1.01(December2013)
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Contents
Chapter1 Introduction 1
1.1ApplicationsLeadingtoDifferentialEquations
1.2 FirstOrderEquations 5
1.3 DirectionFieldsforFirstOrderEquations 16
Chapter2 FirstOrderEquations 30
2.1 LinearFirstOrder Equations 30
2.2 SeparableEquations 45
2.3 ExistenceandUniquenessofSolutionsofNonlinearEquations 55
2.4 TransformationofNonlinearEquationsintoSeparableEquations 63
2.5 ExactEquations 73
2.6 IntegratingFactors 83
Chapter3 NumericalMethods
3.1 Euler’sMethod 96
3.2 TheImprovedEulerMethodandRelatedMethods 109
3.3 TheRunge-KuttaMethod 119
Chapter4 ApplicationsofFirstOrderEquations1em 130
4.1 GrowthandDecay 130
4.2 CoolingandMixing 140
4.3 ElementaryMechanics 151
4.4 AutonomousSecondOrderEquations 162
4.5 ApplicationstoCurves 179
Chapter5 LinearSecondOrderEquations
5.1 HomogeneousLinearEquations 194
5.2 ConstantCoefficientHomogeneousEquations 210
5.3 NonhomgeneousLinearEquations 221
5.4 TheMethodofUndeterminedCoefficientsI 229
iv
5.5 TheMethodofUndeterminedCoefficientsII 238
5.6 ReductionofOrder 248
5.7 VariationofParameters 255
Chapter6 ApplcationsofLinearSecondOrderEquations 268
6.1 SpringProblemsI 268
6.2 SpringProblemsII 279
6.3 TheRLC Circuit 291
6.4 MotionUnderaCentralForce 297
Chapter7 SeriesSolutionsofLinearSecondOrderEquations
7.1 ReviewofPowerSeries 307
7.2 SeriesSolutionsNearanOrdinaryPointI 320
7.3 SeriesSolutionsNearanOrdinaryPointII 335
7.4 RegularSingularPointsEulerEquations 343
7.5 TheMethodofFrobeniusI 348
7.6 TheMethodofFrobeniusII 365
7.7 TheMethodofFrobeniusIII 379
Chapter8 LaplaceTransforms
8.1 IntroductiontotheLaplaceTransform 394
8.2 TheInverseLaplaceTransform 406
8.3 SolutionofInitialValueProblems 414
8.4 TheUnitStepFunction 421
8.5 ConstantCoefficientEquationswithPiecewiseContinuousForcing
Functions 431
8.6 Convolution 441
8.7 ConstantCofficientEquationswithImpulses 453
8.8 ABriefTableofLaplaceTransforms
Chapter9 LinearHigherOrderEquations
9.1 IntroductiontoLinearHigherOrderEquations 466
9.2 HigherOrderConstantCoefficientHomogeneousEquations 476
9.3 UndeterminedCoefficientsforHigherOrderEquations 488
9.4 VariationofParametersforHigherOrderEquations 498
Chapter10 LinearSystemsofDifferentialEquations
10.1 IntroductiontoSystemsofDifferentialEquations 508
10.2 LinearSystemsofDifferentialEquations 516
10.3 BasicTheoryofHomogeneousLinearSystems 522
10.4 ConstantCoefficientHomogeneousSystemsI 530
vi Contents
10.5 ConstantCoefficientHomogeneousSystemsII 543
10.6 ConstantCoefficientHomogeneousSystemsII 557
10.7 VariationofParametersforNonhomogeneousLinearSystems 570
Preface
ElementaryDifferentialEquationswithBoundaryValueProblemsiswrittenforstudentsinscience, en-
gineering,andmathematicswhohavecompletedcalculusthroughpartialdifferentiation.Ifyoursyllabus
includesChapter10(LinearSystemsofDifferentialEquations),yourstudentsshouldhave some prepa-
rationinlinearalgebra.
InwritingthisbookIhavebeenguidedbythetheseprinciples:
An elementary text shouldbe writtenso the student can read itwithcomprehension withouttoo
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muchpain. Ihave triedtoputmyselfinthestudent’splace, andhave chosentoerronthesideof
toomuchdetailratherthannotenough.
Anelementarytextcan’tbebetterthanitsexercises. Thistextincludes1695numberedexercises,
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manywithseveralparts. Theyrangeindifficultyfromroutinetoverychallenging.
Anelementary textshouldbe writteninan informalbutmathematically accurate way, illustrated
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by appropriate graphics. I have triedto formulate mathematical concepts succinctly in language
thatstudentscan understand. Ihave minimizedthenumberofexplicitlystatedtheoremsanddef-
initions, preferring to deal with concepts in a more conversational way, copiously illustratedby
250completelyworkedoutexamples. Whereappropriate,conceptsandresultsaredepictedin144
figures.
AlthoughI believe that the computer is an immensely valuable toolforlearning, doing, and writing
mathematics, theselectionandtreatmentoftopicsinthistextreflects mypedagogicalorientationalong
traditionallines. However, Ihave incorporatedwhatIbelieve tobe thebestuse ofmoderntechnology,
soyoucan selecttheleveloftechnologythatyouwanttoincludeinyourcourse. Thetextincludes336
exercises–identifiedbythesymbols C and C/G –thatcallforgraphicsorcomputationandgraphics.
There are also73laboratoryexercises –identifiedby L –thatrequireextensive useoftechnology. In
addition,severalsectionsincludeinformaladviceontheuseoftechnology.Ifyouprefernottoemphasize
technology,simplyignoretheseexercises andtheadvice.
Therearetwoschoolsofthoughtonwhethertechniquesandapplicationsshouldbetreatedtogetheror
separately. Ihave chosentoseparate them;thus,Chapter2dealswithtechniquesforsolvingfirstorder
equations,andChapter4dealswithapplications. Similarly,Chapter5dealswithtechniquesforsolving
secondorderequations,andChapter6dealswithapplications.However,theexercisesetsofthesections
dealingwithtechniquesincludesomeappliedproblems.
Traditionallyorientedelementarydifferentialequationstextsare occasionallycriticizedas beingcol-
lectionsofunrelatedmethodsforsolvingmiscellaneousproblems. Tosomeextentthisistrue;afterall,
nosinglemethodappliestoallsituations.Nevertheless,Ibelievethatoneideacangoalongwaytoward
unifyingsomeofthetechniquesforsolvingdiverseproblems:variationofparameters. Iusevariationof
parametersattheearliestopportunityinSection2.1,tosolvethenonhomogeneouslinearequation,given
anontrivialsolutionofthecomplementaryequation.Youmayfindthisannoying,sincemostofuslearned
thatoneshoulduseintegratingfactorsforthistask,whileperhapsmentioningthevariationofparameters
optioninanexercise. However,there’slittledifferencebetweenthetwoapproaches,sinceanintegrating
factor isnothingmore thanthe reciprocal of a nontrivialsolutionof the complementary equation. The
advantageofusingvariationofparameters hereisthatitintroducestheconceptinitssimplestformand
vii
viii Preface
focusesthestudent’sattentionontheideaofseekingasolutiony ofadifferentialequationbywritingit
asy uy ,wherey isaknownsolutionofrelatedequationanduisafunctiontobedetermined. Iuse
1 1
D
thisideainnonstandardways,asfollows:
In Section2.4 tosolve nonlinearfirst orderequations, such as Bernoulliequationsand nonlinear
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homogeneousequations.
InChapter3fornumericalsolutionofsemilinearfirstorderequations.
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InSection5.2toavoidthenecessityofintroducingcomplex exponentialsinsolvinga secondor-
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derconstantcoefficienthomogeneousequationwithcharacteristicpolynomialsthathavecomplex
zeros.
In Sections 5.4, 5.5, and 9.3 for the method of undetermined coefficients. (If the method of an-
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nihilatorsis your preferred approach to thisproblem, compare the laborinvolved in solving, for
example,y y y x4ex bythemethodofannihilatorsandthemethodusedinSection5.4.)
00 0
C C D
Introducingvariationofparametersasearlyaspossible(Section2.1)preparesthestudentforthecon-
cept whenitappears againinmore complexforms inSection5.6, where reductionoforderisused not
merelytofindasecondsolutionofthecomplementaryequation,butalsotofindthegeneralsolutionofthe
nonhomogeneousequation,andinSections5.7,9.4,and10.7,thattreattheusualvariationofparameters
problemforsecondandhigherorderlinearequationsandforlinearsystems.
Youmayalsofindthefollowingtobeofinterest:
Section 2.6 deals withintegratingfactors ofthe form (cid:22) p.x/q.y/, inadditionto thoseof the
(cid:15) D
form(cid:22) p.x/and(cid:22) q.y/discussedinmosttexts.
D D
Section4.4makesphase planeanalysisofnonlinearsecondorderautonomousequationsaccessi-
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bletostudentswhohave nottakenlinearalgebra,sinceeigenvaluesandeigenvectorsdonotenter
intothetreatment. Phase plane analysisofconstantcoefficient linearsystems isincludedinSec-
tions10.4-6.
Section4.5presentsanextensivediscussionofapplicationsofdifferentialequationstocurves.
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Section6.4studiesmotionunderacentralforce,whichmaybeusefultostudentsinterestedinthe
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mathematicsofsatelliteorbits.
Sections 7.5-7 present the method of Frobeniusin more detailthan in most texts. The approach
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istosystematize thecomputationsinawaythatavoidsthenecessityofsubstitutingtheunknown
Frobeniusseriesintoeachequation. Thisleadstoefficiencyinthecomputationofthecoefficients
oftheFrobeniussolution.Italsoclarifiesthecasewheretherootsoftheindicialequationdifferby
aninteger(Section7.7).
ThefreeStudentSolutionsManualcontainssolutionsofmostoftheeven-numberedexercises.
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The free Instructor’s Solutions Manual is available by email to [email protected], subject to
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verificationoftherequestor’sfacultystatus.
Thefollowingobservationsmaybehelpfulasyouchooseyoursyllabus:
Section2.3istheonlyspecific prerequisiteforChapter3. Toaccomodate institutionsthatoffera
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separate course innumerical analysis, Chapter3 isnota prerequisiteforany othersection inthe
text.
Preface ix
The sections inChapter4 are independentofeach other, and are notprerequisitesfor any of the
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laterchapters. ThisisalsotrueofthesectionsinChapter6,exceptthatSection6.1isaprerequisite
forSection6.2.
Chapters 7, 8, and 9 can be covered in any order after the topics selected from Chapter 5. For
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example,youcanproceeddirectlyfromChapter5toChapter9.
ThesecondorderEulerequationisdiscussedinSection7.4,whereitsetsthestageforthemethod
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ofFrobenius. AsnotedatthebeginningofSection7.4, ifyouwanttoincludeEulerequationsin
your syllabuswhile omittingthe method of Frobenius, you can skip the introductoryparagraphs
inSection7.4 andbeginwithDefinition7.4.2. Youcan thencover Section7.4immediatelyafter
Section5.2.
WilliamF.Trench
CHAPTER 1
Introduction
INTHISCHAPTERwebeginourstudyofdifferentialequations.
SECTION1.1presentsexamplesofapplicationsthatleadtodifferentialequations.
SECTION1.2introducesbasicconceptsanddefinitionsconcerningdifferentialequations.
SECTION 1.3presentsageometricmethodfordealingwithdifferentialequationsthathasbeen known
for a very longtime, buthas become particularlyusefuland importantwiththe proliferationofreadily
availabledifferentialequationssoftware.
1