Table Of Content“K16435’ — 2017/9/28 — 9:43 — #1
Handbook of
Ordinary Differential Equations
Exact Solutions, Methods, and Problems
“K16435’ — 2017/9/28 — 9:43 — #2
“K16435’ — 2017/9/28 — 9:43 — #3
Handbook of
Ordinary Differential Equations
Exact Solutions, Methods, and Problems
Andrei D. Polyanin
Valentin F. Zaitsev
“K16435’ — 2017/9/28 — 9:43 — #4
MATLAB• is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the
accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB• software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular
use of the MATLAB• software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2018 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
Version Date: 20170928
International Standard Book Number-13: 978-1-4665-6937-9 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity
of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized
in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying,
microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
“K16435’ — 2017/9/28 — 15:05 — #1
CONTENTS
Preface xxiii
Authors xxv
BasicNotationandRemarks xxvii
PartI. Methods forOrdinary Differential Equations 1
1 MethodsforFirst-OrderDifferentialEquations 3
1.1 GeneralConcepts. CauchyProblem. Uniqueness andExistenceTheorems .... 3
1.1.1 Equations SolvedfortheDerivative ................................ 3
1.1.2 Equations NotSolvedfortheDerivative ............................ 7
1.2 Equations SolvedfortheDerivative. SimplestTechniquesofIntegration ...... 8
1.2.1 Equations withSeparableVariablesandRelatedEquations ............ 8
1.2.2 Homogeneous andGeneralized Homogeneous Equations ............. 9
1.2.3 LinearEquationandBernoulli Equation ............................ 10
1.2.4 DarbouxEquationandOtherEquations ............................ 11
1.3 ExactDifferential Equations. Integrating Factor ........................... 12
1.3.1 ExactDifferential Equations ...................................... 12
1.3.2 Integrating Factor ............................................... 13
1.4 RiccatiEquation ...................................................... 13
1.4.1 General Riccati Equation. Simplest Integrable Cases. Polynomial
Solutions ....................................................... 13
1.4.2 UseofParticularSolutionstoConstruct theGeneralSolution ......... 15
1.4.3 SomeTransformations ........................................... 16
1.4.4 SpecialRiccatiEquation ......................................... 17
1.5 AbelEquationsoftheFirstKind ........................................ 18
1.5.1 General FormofAbelEquations oftheFirstKind. SimplestIntegrable
Cases .......................................................... 18
1.5.2 SomeTransformations ........................................... 19
1.6 AbelEquationsoftheSecondKind ...................................... 20
1.6.1 GeneralFormofAbelEquationsoftheSecondKind. SimplestIntegrable
Cases .......................................................... 20
1.6.2 SomeTransformations ........................................... 21
1.6.3 UseofParticular Solutions toConstruct Self-Transformations andthe
GeneralSolution ................................................ 22
1.7 Classification andSpecificFeaturesofSomeClassesofSolutions ............ 25
1.7.1 StableandUnstableSolutions. Equilibrium Points ................... 25
1.7.2 Blow-UpSolutions .............................................. 28
1.7.3 SpaceLocalization ofSolutions ................................... 33
1.7.4 CauchyProblemsAdmittingNon-UniqueSolutions .................. 36
1.8 Equations NotSolvedfortheDerivativeandEquations DefinedParametrically 37
1.8.1 Methodof“Integration byDifferentiation” forEquations NotSolvedfor
theDerivative ................................................... 37
1.8.2 Equations NotSolvedfortheDerivative. SpecificEquations .......... 38
1.8.3 Equations DefinedParametrically andDifferential-Algebraic Equations 40
v
“K16435’ — 2017/9/28 — 15:05 — #2
vi CONTENTS
1.9 ContactTransformations ............................................... 42
1.9.1 GeneralFormofContactTransformations. MethodfortheConstruction
ofContactTransformations ....................................... 42
1.9.2 ExamplesofContactTransformations .............................. 43
1.10 PfaffianEquations .................................................... 44
1.10.1 PreliminaryRemarks .......................................... 44
1.10.2 CompletelyIntegrable PfaffianEquations ......................... 45
1.10.3 PfaffianEquationsNotSatisfying theIntegrability Condition ........ 48
1.11 ApproximateAnalyticMethodsforSolutionofODEs ..................... 49
1.11.1 MethodofSuccessiveApproximations (PicardMethod) ............ 49
1.11.2 Newton–Kantorovich Method ................................... 50
1.11.3 MethodofSeriesExpansion intheIndependent Variable ........... 53
1.11.4 MethodofRegularExpansionintheSmallParameter .............. 56
1.12 DifferentialInequalities andSolutionEstimates .......................... 57
1.12.1 TwoTheoremsonSolutionEstimates ............................ 57
1.12.2 Chaplygin’s TheoremandItsApplications (BilateralEstimatesofthe
CauchyProblemSolution) ...................................... 58
1.13 StandardNumericalMethodsforSolvingOrdinaryDifferential Equations ... 61
1.13.1 Single-StepMethods. Runge–Kutta Methods ..................... 61
1.13.2 Multistep Methods ............................................ 68
1.13.3 Predictor–Corrector Methods ................................... 70
1.13.4 ModifiedMultistep Methods(Butcher’sMethods) ................. 72
1.13.5 StabilityandConvergence ofNumericalMethods .................. 72
1.13.6 Well-andIll-Conditioned Problems .............................. 73
1.14 SpecialNumericalMethods ............................................ 74
1.14.1 SpecialMethodsBasedonAuxiliaryEquations ................... 74
1.14.2 Numerical Integration ofEquations ThatContain FixedSingular
Points ....................................................... 76
1.14.3 Numerical Integration of Equations Defined Parametrically or
Implicitly .................................................... 79
1.14.4 NumericalSolutionofBlow-UpProblems ........................ 81
1.14.5 NumericalSolutionofProblemswithRootSingularity ............. 89
2 MethodsforSecond-OrderLinearDifferential Equations 93
2.1 Homogeneous LinearEquations ......................................... 93
2.1.1 FormulasfortheGeneralSolution. Wronskian Determinant ........... 93
2.1.2 Factorization andSomeTransformations ........................... 95
2.2 Nonhomogeneous LinearEquations ..................................... 96
2.2.1 Existence Theorem. Kummer–LiouvilleTransformation .............. 96
2.2.2 FormulasfortheGeneralSolution ................................. 96
2.3 Representation ofSolutionsasaSeriesintheIndependent Variable .......... 97
2.3.1 Equation Coefficients areRepresentable intheOrdinary PowerSeries
Form .......................................................... 97
2.3.2 EquationCoefficientsHavePolesatSomePoint ..................... 98
2.4 AsymptoticSolutions .................................................. 99
2.4.1 Equations NotContaining y ..................................... 100
x′
2.4.2 Equations Containing y ......................................... 102
x′
“K16435’ — 2017/9/28 — 15:05 — #3
CONTENTS vii
2.5 Boundary ValueProblems. Green’sFunction .............................. 103
2.5.1 First,Second,Third,andSomeOtherBoundaryValueProblems ....... 103
2.5.2 SimplificationofBoundaryConditions. Self-AdjointFormofEquations 106
2.5.3 Green’s andModifiedGreen’sFunctions. Representation Solutionsvia
Green’sorModifiedGreen’sFunctions ............................. 107
2.6 Eigenvalue Problems .................................................. 110
2.6.1 Sturm–Liouville Problem ......................................... 110
2.6.2 GeneralPropertiesoftheSturm–Liouville Problem(2.6.1.1),(2.6.1.2) . 111
2.6.3 ProblemswithBoundaryConditions oftheFirstKind ................ 112
2.6.4 ProblemswithBoundaryConditions oftheSecondKind ............. 113
2.6.5 ProblemswithBoundaryConditions oftheThirdKind ............... 114
2.6.6 ProblemswithMixedBoundaryConditions ......................... 114
2.7 TheoremsonEstimatesandZerosofSolutions ............................ 115
2.7.1 TheoremonEstimatesofSolutions ................................ 115
2.7.2 SturmComparison TheoremonZerosofSolutions .................. 115
2.7.3 QualitativeBehavior ofSolutionsasx ........................ 116
→∞
2.8 NumericalMethods ................................................... 116
2.8.1 Numerov’sMethod(CauchyProblem) ............................. 116
2.8.2 ModifiedShooting Method(Boundary ValueProblems) .............. 117
2.8.3 SweepMethod(Boundary ValueProblems) ......................... 118
2.8.4 MethodofAccelerated Convergence inEigenvalue Problems ......... 119
2.8.5 Well-Conditioned andIll-Conditioned Problems ..................... 120
3 MethodsforSecond-OrderNonlinearDifferentialEquations 123
3.1 GeneralConcepts. CauchyProblem. Uniqueness andExistenceTheorems .... 123
3.1.1 Equations SolvedfortheDerivative. GeneralSolution ................ 123
3.1.2 CauchyProblem. Existence andUniquenessTheorem ................ 123
3.2 SomeTransformations. EquationsAdmittingReductionofOrder ............ 124
3.2.1 Equations NotContaining y orxExplicitly. RelatedEquations ........ 124
3.2.2 Homogeneous Equations ......................................... 125
3.2.3 Generalized Homogeneous Equations .............................. 126
3.2.4 Equations Invariant underScaling–Translation Transformations ....... 126
3.2.5 ExactSecond-Order Equations .................................... 127
3.2.6 Nonlinear EquationsInvolving LinearHomogeneous DifferentialForms 128
3.2.7 Reduction ofQuasilinear EquationstotheNormalForm .............. 129
3.2.8 Equations DefinedParametrically andDifferential-Algebraic Equations 129
3.3 Boundary ValueProblems. Uniqueness andExistenceTheorems. Nonexistence
Theorems ............................................................ 133
3.3.1 Uniqueness andExistenceTheoremsforBoundary ValueProblems .... 134
3.3.2 Reduction ofBoundary ValueProblemstoIntegral Equations. Integral
Identity. Jentzch Theorem ........................................ 137
3.3.3 Theorem onNonexistence ofSolutions tothe FirstBoundary Value
Problem. TheoremsonExistenceofTwoSolutions .................. 139
3.3.4 ExamplesofExistence, Nonuniqueness, andNonexistence ofSolutions
toFirstBoundary ValueProblems ................................. 142
3.3.5 Theorems on Nonexistence of Solutions for the Mixed Problem.
TheoremsonExistenceofTwoSolutions ........................... 145
3.3.6 ExamplesofExistence, Nonuniqueness, andNonexistence ofSolutions
toMixedBoundaryValueProblems ............................... 148
3.3.7 TheoremsonExistence ofTwoSolutionsfortheThirdBoundary Value
Problem ........................................................ 151
“K16435’ — 2017/9/28 — 15:05 — #4
viii CONTENTS
3.3.8 Boundary Value Problems for Linear Equations with Nonlinear
Boundary Conditions ............................................ 152
3.4 MethodofRegularSeriesExpansions withRespecttotheIndependent Variable 154
3.4.1 MethodofExpansion inPowersoftheIndependent Variable .......... 154
3.4.2 Pade´ Approximants .............................................. 156
3.5 MovableSingularities ofSolutionsofOrdinaryDifferential Equations. Painleve´
Equations ............................................................ 157
3.5.1 PreliminaryRemarks. SingularPointsofSolutions .................. 157
3.5.2 FirstPainleve´ Equation .......................................... 158
3.5.3 SecondPainleve´ Equation ........................................ 160
3.5.4 ThirdPainleve´ Equation .......................................... 162
3.5.5 FourthPainleve´ Equation ......................................... 165
3.5.6 FifthPainleve´ Equation .......................................... 167
3.5.7 SixthPainleve´ Equation .......................................... 169
3.6 Perturbation MethodsofMechanicsandPhysics ........................... 171
3.6.1 PreliminaryRemarks. SummaryTableofBasicMethods ............. 171
3.6.2 MethodofRegular(Direct)ExpansioninPowersoftheSmallParameter 173
3.6.3 MethodofScaledParameters(Lindstedt–Poincare´ Method) ........... 174
3.6.4 Averaging Method(VanderPol–Krylov–Bogolyubov Scheme) ........ 175
3.6.5 MethodofTwo-ScaleExpansions (Cole–Kevorkian Scheme) ......... 177
3.6.6 MethodofMatchedAsymptoticExpansions ........................ 178
3.7 GalerkinMethodandItsModifications (Projection Methods) ................ 181
3.7.1 Approximate SolutionforaBoundaryValueProblem ................ 181
3.7.2 GalerkinMethod. GeneralScheme ................................ 182
3.7.3 Bubnov–Galerkin, Moment,andLeastSquaresMethods .............. 182
3.7.4 Collocation Method ............................................. 183
3.7.5 MethodofPartitioning theDomain ................................ 184
3.7.6 LeastSquaredErrorMethod ...................................... 185
3.8 Iteration andNumericalMethods ........................................ 185
3.8.1 MethodofSuccessiveApproximations (CauchyProblem) ............ 185
3.8.2 Runge–Kutta Method(CauchyProblem) ........................... 186
3.8.3 Reduction toaSystemofEquations (CauchyProblem) ............... 186
3.8.4 Predictor–Corrector Methods(CauchyProblem) ..................... 186
3.8.5 ShootingMethod(Boundary ValueProblems) ....................... 187
3.8.6 NumericalMethods forProblemswithEquations DefinedImplicitly or
Parametrically .................................................. 190
3.8.7 NumericalSolutionBlow-UpProblems ............................ 192
4 MethodsforLinearODEsofArbitraryOrder 197
4.1 LinearEquations withConstantCoefficients .............................. 197
4.1.1 Homogeneous LinearEquations. GeneralSolution ................... 197
4.1.2 Nonhomogeneous LinearEquations. GeneralandParticularSolutions .. 198
4.2 LinearEquations withVariableCoefficients ............................... 200
4.2.1 Homogeneous LinearEquations. General Solution. OrderReduction.
LiouvilleFormula ............................................... 200
4.2.2 Nonhomogeneous LinearEquations. General Solution. Superposition
Principle ....................................................... 201
4.2.3 Nonhomogeneous LinearEquations. CauchyProblem. Reduction to
IntegralEquations ............................................... 202
“K16435’ — 2017/9/28 — 15:05 — #5
CONTENTS ix
4.3 LaplaceTransformandtheLaplaceIntegral. Applications toLinearODEs .... 204
4.3.1 LaplaceTransformandtheInverseLaplaceTransform ............... 204
4.3.2 MainProperties oftheLaplaceTransform. Inversion FormulasforSome
Functions ...................................................... 205
4.3.3 LimitTheorems. Representation ofInverse TransformsasConvergent
SeriesandAsymptoticExpansions ................................ 207
4.3.4 Solution oftheCauchyProblem forConstant-Coefficient LinearODEs.
Applications toIntegro-Differential Equations ...................... 209
4.3.5 Solution ofLinearEquations withPolynomial Coefficients Usingthe
LaplaceTransform .............................................. 211
4.3.6 Solution ofLinearEquations withPolynomial Coefficients Usingthe
LaplaceIntegral ................................................. 212
4.4 AsymptoticSolutions ofLinearEquations ................................ 213
4.4.1 Fourth-OrderLinearDifferential Equations ......................... 213
4.4.2 Higher-Order LinearDifferential Equations ......................... 214
4.5 Collocation Method ................................................... 215
4.5.1 StatementoftheProblem. ApproximateSolution .................... 215
4.5.2 Convergence Theorem ........................................... 215
5 MethodsforNonlinearODEsofArbitraryOrder 217
5.1 GeneralConcepts. CauchyProblem. Uniqueness andExistenceTheorems .... 217
5.1.1 Equations SolvedfortheDerivative. GeneralSolution ................ 217
5.1.2 SomeTransformations ........................................... 218
5.2 Equations AdmittingReduction ofOrder ................................. 218
5.2.1 Equations NotContaining y orxExplicitly ......................... 218
5.2.2 Homogeneous Equations ......................................... 219
5.2.3 Generalized Homogeneous Equations .............................. 220
5.2.4 Equations Invariant underScaling-Translation Transformations ........ 220
5.2.5 OtherEquations ................................................. 220
5.3 MethodforConstruction ofSolvableEquationsofGeneralForm ............ 222
5.3.1 Description oftheMethod ........................................ 222
5.3.2 Illustrative Examples ............................................ 223
5.4 NumericalIntegration ofn-orderEquations ............................... 224
5.4.1 NumericalSolutionoftheCauchyProblemforn-orderODEs ......... 224
5.4.2 NumericalSolutionofEquations DefinedImplicitlyorParametrically .. 224
6 MethodsforLinearSystemsofODEs 227
6.1 SystemsofLinearConstant-Coefficient Equations ......................... 227
6.1.1 Systems ofFirst-Order Linear Homogeneous Equations. General
Solution ........................................................ 227
6.1.2 Systems ofFirst-Order Linear Homogeneous Equations. Particular
Solutions ....................................................... 228
6.1.3 Nonhomogeneous SystemsofLinearFirst-OrderEquations ........... 230
6.1.4 Homogeneous LinearSystemsofHigher-OrderDifferential Equations . 231
6.1.5 NormalCoordinates andNaturalOscillations ....................... 232
6.1.6 Nonhomogeneous Higher-OrderLinearSystems. D’Alembert’sMethod 233
6.1.7 Usage of the Laplace Transform for Solving Linear Systems of
Equations ...................................................... 234
6.1.8 Classification of Equilibrium Points of Two-Dimensional Linear
Systems ........................................................ 235
