Elementary differential equations

784 Pages·2008·8.73 MB·English

by  Edwards, Charles Henry;Penney, David E.;Calvis, David

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Sixth Edition
C. Henry Edwards
David E. Penney
The University of Georgia
with the assistance of
David Calvis
Baldwin-Wallace College
Upper Saddle River, NJ 07458
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Preface vii
CHAPTER First-Order Differential Equations 1
11
1.1 DifferentialEquationsandMathematicalModels 1
1.2 IntegralsasGeneralandParticularSolutions 10
1.3 SlopeFieldsandSolutionCurves 19
1.4 SeparableEquationsandApplications 32
1.5 LinearFirst-OrderEquations 46
1.6 SubstitutionMethodsandExactEquations 59
1.7 PopulationModels 74
1.8 Acceleration-VelocityModels 85
CHAPTER Linear Equations of Higher Order 100
22
2.1 Introduction: Second-OrderLinearEquations 100
2.2 GeneralSolutionsofLinearEquations 113
2.3 HomogeneousEquationswithConstantCoefficients 124
2.4 MechanicalVibrations 135
2.5 NonhomogeneousEquationsandUndeterminedCoefficients 148
2.6 ForcedOscillationsandResonance 162
2.7 ElectricalCircuits 173
2.8 EndpointProblemsandEigenvalues 180
CHAPTER Power Series Methods 194
33
3.1 IntroductionandReviewofPowerSeries 194
3.2 SeriesSolutionsNearOrdinaryPoints 207
3.3 RegularSingularPoints 218
3.4 MethodofFrobenius: TheExceptionalCases 233
3.5 Bessel'sEquation 248
3.6 ApplicationsofBesselFunctions 257
iii
iv Contents
CHAPTER Laplace Transform Methods 266
44
4.1 LaplaceTransformsandInverseTransforms 266
4.2 TransformationofInitialValueProblems 277
4.3 TranslationandPartialFractions 289
4.4 Derivatives,Integrals,andProductsofTransforms 297
4.5 PeriodicandPiecewiseContinuousInputFunctions 304
4.6 ImpulsesandDeltaFunctions 316
CHAPTER Linear Systems of Differential Equations 326
55
5.1 First-OrderSystemsandApplications 326
5.2 TheMethodofElimination 338
5.3 MatricesandLinearSystems 347
5.4 TheEigenvalueMethodforHomogeneousSystems 366
5.5 Second-OrderSystemsandMechanicalApplications 381
5.6 MultipleEigenvalueSolutions 393
5.7 MatrixExponentialsandLinearSystems 407
5.8 NonhomogeneousLinearSystems 420
CHAPTER Numerical Methods 430
66
6.1 NumericalApproximation: Euler'sMethod 430
6.2 ACloserLookattheEulerMethod 442
6.3 TheRunge-KuttaMethod 453
6.4 NumericalMethodsforSystems 464
CHAPTER Nonlinear Systems and Phenomena 480
77
7.1 EquilibriumSolutionsandStability 480
7.2 StabilityandthePhasePlane 488
7.3 LinearandAlmostLinearSystems 500
7.4 EcologicalModels: PredatorsandCompetitors 513
7.5 NonlinearMechanicalSystems 526
7.6 ChaosinDynamicalSystems 542
CHAPTER Fourier Series Methods 554
88
8.1 PeriodicFunctionsandTrigonometricSeries 554
8.2 GeneralFourierSeriesandConvergence 563
8.3 FourierSineandCosineSeries 570
8.4 ApplicationsofFourierSeries 581
Contents v
8.5 HeatConductionandSeparationofVariables 586
8.6 VibratingStringsandtheOne-DimensionalWaveEquation 599
8.7 Steady-StateTemperatureandLaplace'sEquation 611
CHAPTER Eigenvalues and Boundary Value Problems 622
99
9.1 Sturm-LiouvilleProblemsandEigenfunctionExpansions 622
9.2 ApplicationsofEigenfunctionSeries 633
9.3 SteadyPeriodicSolutionsandNaturalFrequencies 642
9.4 CylindricalCoordinateProblems 650
9.5 Higher-DimensionalPhenomena 664
References for Further Study 683
Appendix: Existence and Uniqueness of Solutions 687
Answers to Selected Problems 701
Index I-1
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The evolution of the present text in successive editions is based on experience
teaching the introductory differential equations course with an emphasis on
conceptual ideas and the use of applications and projects to involve students in
active problem-solving experiences. At various points our approach reflects the
widespreaduseoftechnicalcomputingenvironmentslikeMaple,Mathematica,and
MATLAB for the graphical, numerical, or symbolic solution of differential equa-
tions. Nevertheless,wecontinuetobelievethatthetraditionalelementaryanalytical
methodsofsolutionareimportantforstudentstolearnanduse. Onereasonisthat
effective and reliable use of computer methods often requires preliminary analysis
using standard symbolic techniques; the construction of a realistic computational
modeloftenisbasedonthestudyofasimpleranalyticalmodel.
Principal Features of This Revision
Whilethesuccessfulfeaturesofprecedingeditionshavebeenretained,theexposi-
tionhasbeensignificantlyenhancedineverychapterandinmostindividualsections
of the text. Both new graphics and new text have been inserted where needed for
improved student understanding of key concepts. However, the solid class-tested
chapter and section structure of the book is unchanged, so class notes and syllabi
willnotrequirerevisionforuseofthisnewedition. Thefollowingexamplesofthis
revision illustrate the way the local structure of the text has been augmented and
polishedforthisedition.
Chapter1: NewFigures1.3.9and1.3.10showingdirectionfieldsthatillus-
tratefailureofexistenceanduniquenessofsolutions(page24);newProblems
34 and 35 showing that small changes in initial conditions can make big dif-
ferencesinresults,butbigchangesininitialconditionsmaysometimesmake
onlysmalldifferencesinresults(page30);newRemarks1and2clarifyingthe
concept of implicit solutions (page 35); new Remark clarifying the meaning
ofhomogeneityforfirst-orderequations(page61);additionaldetailsinserted
in the derivation of the rocket propulsion equation (page 95), and new Prob-
lem 5 inserted to investigate the liftoff pause of a rocket on the launch pad
sometimesobservedbeforeblastoff(page97).
Chapter 2: New explanation of signs and directions of internal forces in
mass-spring systems (page 101); new introduction of differential operators
and clarification of the algebra of polynomial operators (page 127); new in-
troduction and illustration of polar exponential forms of complex numbers
(page132); fullerexplanationofmethodofundeterminedcoefficientsinEx-
amples1and3(page149–150);newRemarks1and2introducing“shooting”
vii
viii Preface
terminology,andnewFigures2.8.1and2.8.2illustratingwhysomeendpoint
valueproblemshaveinfinitelymanysolutions,whileothershavenosolutions
at all (page 181); new Figures 2.8.4 and 2.8.5 illustrating different types of
eigenfunctions(pages183–184).
Chapter 3: New Problem 35 on determination of radii of convergence of
power series solutions of differential equations (page 218); new Example 3
justbeforethesubsectiononlogarithmiccasesinthemethodofFrobenius,to
illustratefirstthereduction-of-orderformulawithasimplenon-seriesproblem
(page239).
Chapter4: Newdiscussionclarifyingfunctionsofexponentialorderandex-
istence of Laplace transforms (page 273); new Remark discussing the me-
chanics of partial-fraction decomposition (page 279); new much-expanded
discussion of the proof of the Laplace-transform existence theorem and its
extension to include the jump discontinuities that play an important role in
manypracticalapplications(page286–287).
Chapter 5: New Problems 20–23 for student exploration of three-railway-
carssystemswithdifferentinitialvelocityconditions(page392);newRemark
illustrating the relation between matrix exponential methods and the gener-
alized eigenvalue methods discussed previously (page 416); new exposition
inserted at end of section to explain the connection between matrix variation
ofparametershereand(scalar)variationofparametersforsecond-orderequa-
tionsdiscussedpreviouslyinChapter3(page427).
Chapter 6: New discussion with new Figures 6.3.11 and 6.3.12 clarifying
thedifferencebetweenrotatingandnon-rotatingcoordinatesystemsinmoon-
earthorbitproblems(page473).
Chapter 7: New remarks on phase plane portraits, autonomous systems,
and critical points (page 488-490); new introduction of linearized systems
(page 502); new 3-dimensional Figures 6.5.18 and 6.5.20 illustrating Lorenz
andRo¨sslertrajectories(page552–553).
Chapter 8: New considerably expanded explanation of even and odd exten-
sions and their Fourier sine-cosine series (page 572); new discussion of pe-
riodic and non-periodic particular solutions illustrated by new Figure 8.4.4,
togetherwithnewProblems19and20atendofsection(pages582,586);new
examplediscussioninsertedatendofsectiontoillustratetheeffectsofdamp-
inginmass-springsystems(page585);newdiscussionofsignsanddirection
ofheatflowinthederivationoftheheatequation(page587).
Chapter 9: Clarification of the effect of internal stretching in deriving the
wave equation for longitudinal vibrations of a bar (page 635–636); new Fig-
ures9.5.15and9.5.16illustratingoceanwavesonasmallplanet(page681).
Throughout the text, almost 700 computer-generated figures show students
vividpicturesofdirectionfields,solutioncurves,andphaseplaneportraitsthatbring
symbolicsolutionsofdifferentialequationstolife.
About15applicationmodulesfollowkeysectionsthroughoutthetext. Their
purposeistoaddconcreteappliedemphasisandtoengagestudentsismoreexten-
siveinvestigationsthanaffordedbytypicalexercisesandproblems.
A solid numerical emphasis provided where appropriate (as in Chapter 6 on
NumericalMethods)bytheinclusionofgenericnumericalalgorithmsandalimited
numberofillustrativegraphingcalculator,BASIC,andMATLABroutines.
Preface ix
Organization and Content
Thetraditionalorganizationofthistextstillaccommodatesfreshnewmaterialand
combinationsoftopics. Forinstance:
• ThefinaltwosectionsofChapter1(onpopulationsandelementarymechan-
ics)offeranearlyintroductiontomathematicalmodelingwithsignificantap-
plications.
• The final section of Chapter 2 offers unusually early exposure to endpoint
problems and eigenvalues, with interesting applications to whirling strings
andbuckledbeams.
• Chapter3combinesacompleteandsolidtreatmentofinfiniteseriesmethods
withinterestingapplicationsofBesselfunctionsinitsfinalsection.
• Chapter 4 combines a complete and solid treatment of Laplace transform
methods with brief coverage of delta functions and their applications in its
finalsection.
• Chapter 5 provides an unusually flexible treatment of linear systems. Sec-
tions 5.1 and 5.2 offer an early, intuitive introduction to first-order systems
and models. The chapter continues with a self-contained treatment of the
necessary linear algebra, and then presents the eigenvalue approach to linear
systems. It includes an unusual number of applications (ranging from brine
tanks to railway cars) of all the various cases of the eigenvalue method. The
coverageofexponentialmatricesinSection5.7isexpandedfromearlieredi-
tions.
• Chapter 6 on numerical methods begins in Section 6.1 with the elementary
Euler method for single equations and ends in Section 6.4 with the Runge-
Kuttamethodforsystemsandapplicationstoorbitsofcometsandsatellites.
• Chapter7onnonlinearsystemsandphenomenarangesfromphaseplaneanal-
ysistoecologicalandmechanicalsystemstoaninnovativeconcludingsection
on chaos and bifurcation in dynamical systems. Section 7.6 presents an ele-
mentary introduction to such contemporary topics as period-doubling in bio-
logicalandmechanicalsystems,thepitchforkdiagram,andtheLorenzstrange
attractor(allillustratedwithvividcomputergraphics).
• Chapter8coversFourierserieswithapplicationstosolutionoftheheat,wave,
andLaplaceequations.
• Chapter9beginswithSturm-LiouvilleODEproblemsandends(inSection9.5)
withsomesubstantialexamplesofhigher-dimensionalPDEphenomena.
Thisbookincludesadequatematerialfordifferentintroductorycoursesvary-
ing in length from one quarter to two semesters. The briefer version, Elementary
DifferentialEquations(0-13-239730-7),endswithChapter7onnonlinearsystems
and phenomena (and thus omits the material on Fourier series, separation of vari-
ables,andpartialdifferentialequations).
Applications
To sample the range of applications in this text, take a look at the following ques-
tions: