Ordinary Differential Equations: An Introduction to the Fundamentals (Textbooks in Mathematics)

907 Pages·2019·13.983 MB·English

by  USA) Howell, Kenneth B. (The University of Alabama in Huntsville

ORDINARY
DIFFERENTIAL
EQUATIONS

An Introduction to the Fundamentals

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

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Jonathan L. Gross, Jay Yellen and Mark Anderson

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SECOND EDITION
Steven G. Krantz

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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