Ordinary differential equations : an introduction to the fundamentals

907 Pages·2019·13.978 MB·English

by  Howell, Kenneth B.

ORDINARY
DIFFERENTIAL
EQUATIONS

An Introduction to the Fundamentals

Second Edition
Textbooks in Mathematics
Series editors:
Al Boggess and Ken Rosen

CRYPTOGRAPHY: THEORY AND PRACTICE, FOURTH EDITION
Douglas R. Stinson and Maura B. Paterson

GRAPH THEORY AND ITS APPLICATIONS, THIRD EDITION
Jonathan L. Gross, Jay Yellen and Mark Anderson

COMPLEX VARIABLES: A PHYSICAL APPROACH WITH APPLICATIONS,
SECOND EDITION
Steven G. Krantz

GAME THEORY: A MODELING APPROACH
Richard Alan Gillman and David Housman

FORMAL METHODS IN COMPUTER SCIENCE
Jiacun Wang and William Tepfenhart

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Marshall A. Whittlesey

AN INTRODUCTION TO MATHEMATICAL PROOFS
Nicholas A. Loehr

COMPUTATIONAL PARTIAL DIFFERENTIAL EQUATIONS USING MATLAB®
Jichun Li and Yi-Tung Chen

AN ELEMENTARY TRANSITION TO ABSTRACT MATHEMATICS
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MATHEMATICAL MODELING WITH EXCEL, SECOND EDITION
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PRINCIPLES OF FOURIER ANALYSIS, SECOND EDITION
Kenneth B. Howell

https://www.crcpress.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH
ORDINARY
DIFFERENTIAL
EQUATIONS

An Introduction to the Fundamentals

Second Edition

Kenneth B. Howell
University of Alabama in Huntsville, USA
Second edition published 2020
by CRC Press
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and by CRC Press
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[First edition published by CRC Press 2015]

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Typeset in NimbusRomNo9L
by Author
Contents
Preface (With ImportantInformation fortheReader)
I TheBasics 1
1 TheStartingPoint: BasicConceptsandTerminology 3
1.1 DifferentialEquations: BasicDefinitionsandClassifications . . . . . . . . . . . . 3
1.2 WhyCareAboutDifferentialEquations? SomeIllustrativeExamples . . . . . . . 8
1.3 MoreonSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 IntegrationandDifferentialEquations 21
2.1 Directly-IntegrableEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 OnUsingIndefiniteIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 OnUsingDefiniteIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 IntegralsofPiecewise-DefinedFunctions . . . . . . . . . . . . . . . . . . . . . . 28
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
II First-Order Equations 35
3 SomeBasicsaboutFirst-OrderEquations 37
3.1 AlgebraicallySolvingfortheDerivative . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Constant(orEquilibrium)Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 OntheExistenceandUniquenessofSolutions . . . . . . . . . . . . . . . . . . . . 42
3.4 ConfirmingtheExistenceofSolutions(CoreIdeas) . . . . . . . . . . . . . . . . . 44
3.5 DetailsintheProofofTheorem3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 OnProvingTheorem3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Appendix: ALittleMultivariableCalculus . . . . . . . . . . . . . . . . . . . . . . 59
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 SeparableFirst-OrderEquations 65
4.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 ConstantSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 ExplicitVersusImplicitSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 FullProcedureforSolvingSeparableEquations . . . . . . . . . . . . . . . . . . . 77
4.5 Existence,Uniqueness,andFalseSolutions . . . . . . . . . . . . . . . . . . . . . 78
4.6 OntheNatureofSolutionstoDifferentialEquations. . . . . . . . . . . . . . . . . 81
4.7 UsingandGraphingImplicitSolutions . . . . . . . . . . . . . . . . . . . . . . . . 83
4.8 OnUsingDefiniteIntegralswithSeparableEquations . . . . . . . . . . . . . . . . 88
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
v
vi
5 LinearFirst-OrderEquations 93
5.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 SolvingFirst-OrderLinearEquations . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 OnUsingDefiniteIntegralswithLinearEquations . . . . . . . . . . . . . . . . . 100
5.4 Integrability,ExistenceandUniqueness . . . . . . . . . . . . . . . . . . . . . . . 102
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 SimplifyingThroughSubstitution 105
6.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 LinearSubstitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 HomogeneousEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4 BernoulliEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 TheExactFormandGeneralIntegratingFactors 117
7.1 TheChainRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 TheExactForm,Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.3 SolvingEquationsinExactForm . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.4 TestingforExactness—PartI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5 “ExactEquations”: ASummary . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.6 ConvertingEquationstoExactForm . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.7 TestingforExactness—PartII . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 ReviewExercisesforPartofPartII 143
9 SlopeFields: GraphingSolutionsWithouttheSolutions 145
9.1 MotivationandBasicConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 TheBasicProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 ObservingLong-TermBehaviorinSlopeFields . . . . . . . . . . . . . . . . . . . 152
9.4 ProblemPointsinSlopeFields,andIssuesofExistenceandUniqueness . . . . . . 158
9.5 TestsforStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
10 NumericalMethodsI:TheEulerMethod 177
10.1 DerivingtheStepsoftheMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.2 ComputingviatheEulerMethod(Illustrated) . . . . . . . . . . . . . . . . . . . . 180
10.3 UsingtheResultsoftheMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.4 ReducingtheError . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
10.5 ErrorAnalysisfortheEulerMethod . . . . . . . . . . . . . . . . . . . . . . . . . 187
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
11 TheArtandScienceofModelingwithFirst-OrderEquations 197
11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.2 ARabbitRanch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11.3 ExponentialGrowthandDecay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.4 TheRabbitRanch,Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
11.5 NotesontheArtandScienceofModeling . . . . . . . . . . . . . . . . . . . . . . 207
11.6 MixingProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.7 SimpleThermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
vii
12 NumericalMethodsII:BeyondtheEulerMethod 221
12.1 ForwardandBackwardEulerMethods . . . . . . . . . . . . . . . . . . . . . . . . 221
12.2 TheImprovedEulerMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.3 AFewOtherMethodsWorthBriefDiscussion . . . . . . . . . . . . . . . . . . . . 230
12.4 TheClassicRunge-KuttaMethod . . . . . . . . . . . . . . . . . . . . . . . . . . 232
12.5 SomeAdditionalComments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
III Second-andHigher-Order Equations 243
13 Higher-OrderEquations: ExtendingFirst-OrderConcepts 245
13.1 TreatingSomeSecond-OrderEquationsasFirst-Order . . . . . . . . . . . . . . . 246
13.2 TheOtherClassofSecond-OrderEquations“EasilyReduced”toFirst-Order . . . 250
13.3 Initial-ValueProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
13.4 OntheExistenceandUniquenessofSolutions . . . . . . . . . . . . . . . . . . . . 256
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
14 Higher-OrderLinearEquationsandtheReductionofOrderMethod 263
14.1 LinearDifferentialEquationsofAllOrders . . . . . . . . . . . . . . . . . . . . . 263
14.2 IntroductiontotheReductionofOrderMethod . . . . . . . . . . . . . . . . . . . 266
14.3 ReductionofOrderforHomogeneousLinearSecond-OrderEquations . . . . . . . 267
14.4 ReductionofOrderforNonhomogeneousLinearSecond-OrderEquations . . . . . 272
14.5 ReductionofOrderinGeneral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
15 GeneralSolutionstoHomogeneousLinearDifferentialEquations 279
15.1 Second-OrderEquations(Mainly) . . . . . . . . . . . . . . . . . . . . . . . . . . 279
15.2 HomogeneousLinearEquationsofArbitraryOrder . . . . . . . . . . . . . . . . . 290
15.3 LinearIndependenceandWronskians . . . . . . . . . . . . . . . . . . . . . . . . 291
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
16 VerifyingtheBigTheoremsandanIntroductiontoDifferentialOperators 299
16.1 VerifyingtheBigTheoremonSecond-Order,HomogeneousEquations . . . . . . . 299
16.2 ProvingtheMoreGeneralTheoremsonGeneralSolutionsandWronskians . . . . 306
16.3 LinearDifferentialOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
17 Second-OrderHomogeneousLinearEquationswithConstantCoefficients 317
17.1 DerivingtheBasicApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
17.2 TheBasicApproach,Summarized . . . . . . . . . . . . . . . . . . . . . . . . . . 320
17.3 Case1: TwoDistinctRealRoots . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
17.4 Case2: OnlyOneRoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
17.5 Case3: ComplexRoots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
17.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
18 Springs: PartI 337
18.1 ModelingtheAction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
18.2 TheMass/SpringEquationandItsSolutions . . . . . . . . . . . . . . . . . . . . . 341
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
viii
19 ArbitraryHomogeneousLinearEquationswithConstantCoefficients 353
19.1 SomeAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
19.2 SolvingtheDifferentialEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
19.3 MoreExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
19.4 OnVerifyingTheorem19.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
19.5 OnVerifyingTheorem19.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
20 EulerEquations 371
20.1 Second-OrderEulerEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
20.2 TheSpecialCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
20.3 EulerEquationsofAnyOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
20.4 TheRelationBetweenEulerandConstantCoefficientEquations . . . . . . . . . . 381
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
21 NonhomogeneousEquationsinGeneral 385
21.1 GeneralSolutionstoNonhomogeneousEquations . . . . . . . . . . . . . . . . . . 385
21.2 SuperpositionforNonhomogeneousEquations . . . . . . . . . . . . . . . . . . . 389
21.3 ReductionofOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
22 MethodofUndeterminedCoefficients(aka: MethodofEducatedGuess) 395
22.1 BasicIdeas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
22.2 GoodFirstGuessesforVariousChoicesof g . . . . . . . . . . . . . . . . . . . . 398
22.3 WhentheFirstGuessFails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
22.4 MethodofGuessinGeneral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
22.5 CommonMistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
22.6 UsingthePrincipleofSuperposition . . . . . . . . . . . . . . . . . . . . . . . . . 408
22.7 OnVerifyingTheorem22.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
23 Springs: PartII(ForcedVibrations) 415
23.1 TheMass/SpringSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
23.2 ConstantForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
23.3 ResonanceandSinusoidalForces. . . . . . . . . . . . . . . . . . . . . . . . . . . 418
23.4 MoreonUndampedMotionunderNonresonantSinusoidalForces . . . . . . . . . 424
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
24 VariationofParameters(ABetterReductionofOrderMethod) 431
24.1 Second-OrderVariationofParameters . . . . . . . . . . . . . . . . . . . . . . . . 431
24.2 VariationofParametersforEvenHigherOrderEquations . . . . . . . . . . . . . . 439
24.3 TheVariationofParametersFormula . . . . . . . . . . . . . . . . . . . . . . . . . 442
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
25 ReviewExercisesforPartIII 447
IV TheLaplaceTransform 449
26 TheLaplaceTransform(Intro) 451
26.1 BasicDefinitionandExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
26.2 LinearityandSomeMoreBasicTransforms . . . . . . . . . . . . . . . . . . . . . 457
26.3 TablesandaFewMoreTransforms. . . . . . . . . . . . . . . . . . . . . . . . . . 459
ix
26.4 TheFirstTranslationIdentity(andMoreTransforms) . . . . . . . . . . . . . . . . 464
26.5 WhatIs“LaplaceTransformable”? (andSomeStandardTerminology) . . . . . . . 466
26.6 FurtherNotesonPiecewiseContinuityandExponentialOrder . . . . . . . . . . . 471
26.7 ProvingTheorem26.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
27 DifferentiationandtheLaplaceTransform 481
27.1 TransformsofDerivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
27.2 DerivativesofTransforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
27.3 TransformsofIntegralsandIntegralsofTransforms . . . . . . . . . . . . . . . . . 488
27.4 Appendix: DifferentiatingtheTransform. . . . . . . . . . . . . . . . . . . . . . . 493
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
28 TheInverseLaplaceTransform 499
28.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
28.2 LinearityandUsingPartialFractions . . . . . . . . . . . . . . . . . . . . . . . . . 501
28.3 InverseTransformsofShiftedFunctions . . . . . . . . . . . . . . . . . . . . . . . 507
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
29 Convolution 511
29.1 Convolution: TheBasics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
29.2 ConvolutionandProductsofTransforms . . . . . . . . . . . . . . . . . . . . . . . 515
29.3 ConvolutionandDifferentialEquations(Duhamel’sPrinciple) . . . . . . . . . . . 519
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
30 Piecewise-DefinedFunctionsandPeriodicFunctions 525
30.1 Piecewise-DefinedFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
30.2 The“TranslationAlongtheT-Axis”Identity . . . . . . . . . . . . . . . . . . . . 528
30.3 RectangleFunctionsandTransformsofMorePiecewise-DefinedFunctions . . . . 533
30.4 ConvolutionwithPiecewise-DefinedFunctions . . . . . . . . . . . . . . . . . . . 537
30.5 PeriodicFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
30.6 AnExpandedTableofIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
30.7 Duhamel’sPrincipleandResonance . . . . . . . . . . . . . . . . . . . . . . . . . 546
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
31 DeltaFunctions 557
31.1 VisualizingDeltaFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
31.2 DeltaFunctionsinModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
31.3 TheMathematicsofDeltaFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 562
31.4 DeltaFunctionsandDuhamel’sPrinciple . . . . . . . . . . . . . . . . . . . . . . 566
31.5 Some“Issues”withDeltaFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 568
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
V PowerSeries andModified PowerSeries Solutions 575
32 SeriesSolutions: Preliminaries 577
32.1 InfiniteSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
32.2 PowerSeriesandAnalyticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 582
32.3 ElementaryComplexAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
32.4 AdditionalBasicMaterialThatMayBeUseful . . . . . . . . . . . . . . . . . . . 594
AdditionalExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599