Table Of ContentOrdinary differential equations—the building blocks ofmathematical model-
ing—are also key elements ofdisciplines as diverse as engineering and eco-
nomics. Although mastery of these equations is essential, adhering to any
onemethodofsolvingthemisnot: Thisbookstressesalternativeexamplesand
analysesbymeansofwhichthestudentcanbuildanunderstandingofanumber
ofapproachestofindingsolutionsandunderstandingtheirbehavior.
The text includes briefexpositions ofstandard topics, including first-order
equations,homogeneousandnonhomogeneous second-orderlinearequations,
power series expansions aboutregularand regular singularpoints, linearsys-
temstheory, and stabilityconceptsforboththephaseplaneandhigher-dimen-
sional systems. A variety ofexercises and examples is included, and readers
areencouragedtotryalternativeapproachestofindsolutionsthatintegrateand
build upon ideas introduced in earlierchapters. This book offers not only an
appliedperspectiveforthe studentlearningto solvedifferential equations, but
alsothechallengetoapplytheseanalyticaltoolsinthecontextofsingularper-
turbations, which arises in many areas ofapplication. An important resource
for the advanced undergraduate, this book would be equally useful for the
beginning graduate studentinvestigating furtherapproaches to these essential
equations.
Thinking About Ordinary Differential Equations
Cambridge Texts inAppliedMathematics
MANAGING EDITOR
ProfessorD.G.Crighton,DepartmentofAppliedMathematicsandTheoreticalPhysics,
UniversityofCambridge,UK.
EDITORIAL BOARD
ProfessorMJ.Ablowitz,PrograminAppliedMathematics,
UniversityofColorado,Boulder,USA.
ProfessorJ.-L.Lions,CollegedeFrance,France.
ProfessorA.Majda,DepartmentofMathematics,NewYorkUniversity,USA.
Dr. J.Ockendon,CentreforIndustrialandAppliedMathematics,UniversityofOxford,UK.
ProfessorEB.Saff,DepartmentofMathematics,UniversityofSouthFlorida,USA.
MaximumandMinimumPrinciples
MJ.Sewell
Solitons
P.G.DrazinandRS.Johnson
TheKinematicsofMixing
J.M. Ottino
IntroductiontoNumericalLinearAlgebraandOptimisation
PhillippeG. Ciarlet
IntegralEquations
DavidPorterandDavidS.G.Stirling
PerturbationMethods
EJ.Hinch
TheThermomechanicsofPlasticityandFracture
GerardA.Maugin
BoundaryIntegralandSingularityMethodsforLinearizedViscousFlow
C.Pozrikidis
NonlinearSystems
RG.Drazin
Stability,InstabilityandChaos
PaulGlendinning
AppliedAnalysisoftheNavier—StokesEquations
C.R.DoeringandJD.Gibbon
ViscousFlow
H. OckendonandJR. Ockendon
Similarity,Self-similarityandIntermediateAsymptotics
G.I.Barenblatt
AFirstCourseintheNumericalAnalysisofDifferentialEquations
A.[series
ComplexVariables: IntroductionandApplications
MarkJ.AblowitzandAthanssiosS.Fokas
ThinkingAboutOrdinaryDifferentialEquations
RobertE. O’Mallqv,Jr.
ThinkingAbout Ordinary Difl’erential Equations
ROBERT E. O’MALLEY, JR.
UniversityofWashington
3‘3 CAMBRIDGE
UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
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CAMBRIDGE UNIVERSITY PRESS
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©CambridgeUniversityPress 1997
Thisbookisincopyright. Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished1997
TypesetinTimesRoman
LibraryofCongressCaraloging-in-PublicationData
O’Malley,RobertE,
Thinkingaboutordinarydifferentialequations/
RobertE.O’Malley,Jr.
p. cm.—(Cambridgetextsinappliedmathematics)
Includesbibliographicalreference(p.243)andindex.
ISBN0-521-55314—8(hardback). -ISBN0-521-55742-9(pbk.)
1. Differentialequations. 1.Title. H.Series.
QA372.057 1997
515’.352—dc20 96-14825
CIP
Acataloguerecordforthisbookisavailablefrom
theBritishLibrary
ISBN 0-521-55314-8hardback
ISBN 0-521-55742-9paperback
Transferredtodigitalprinting2004
To Candy, Patrick, Hmothy, andDaniel ...
GreatO’Malleys, blissfullyunenlightenedconcerning
difierentialequations
Contents
Preface pageix
First-orderequations 1
1.1 Introductoryremarks 1
1.2 Separableequations 13
1.3 Exactequations 16
1.4 Linearequations 20
1.5 Equations ofhomogeneoustype 24
1.6 Special second-orderequations 25
1.7 Epilog 28
1.8 Exercises 28
Linearsecond-orderequations 42
2.1 Homogeneousequations 42
2.2 NonhomogeneousEquations 53
2.3 Applications 61
2.4 Exercises 65
Powerseriessolutionsandspecialfunctions 80
3.1 Taylorseries 80
3.2 Regularsingularpoints andthemethodofFrobenius 88
3.3 Besselfunctions 99
3.4 LegendrepolynomialsandSturm-Liouvilleproblems 106
3.5 Earthquakeprotection 111
3.6 Exercises 113
Systemsoflineardifferentialequations 133
Thefundamentalmatrix 133
Vii
viii Contents
4.2 Thematrixexponential 153
4.3 Exercises 161
Stabilityconcepts 176
5.1 Two—dimensionallinearsystems 176
5.2 Usingthephaseplanefornonlinearproblems l84
5.3 Stabilityforhigher—dimensional systems 190
5.4 Liapunovfunctions 196
5.5 Exercises 198
Singularperturbationmethods 208
6.1 Regularandsingularperturbations 208
6.2 Linearinitial valueproblems 212
6.3 Nonlinearinitial valueproblems 217
6.4 Two-pointproblems 223
6.5 Acombustionmodelexhibitingmetastability 233
6.6 Exercises 239
Referenceson singularperturbations
GeneralReferences 243
Index 244
Preface
This small book is intended for use by students in the applied sciences and engi-
neering who already have some elementary knowledge of ordinary differential
equations. It aims to emphasize the variety of analytical approaches available
and to teach simple techniques to use in their own technical work and in under-
standing the behavior of solutions to many standard problems. The exercises at
the end of each chapter, in particular, are intended to be the primary learning tool,
so fairly detailed solutions are provided for many of them. The important job of
interpreting solutions in their underlying physical context is left to the reader.
Good calculus skills are called for. Some familiarity with numerical and/or
symbolic computing and with matrix analysis would also be helpful, but is
not necessary. We will not hesitate to introduce needed theory, without proof,
in order to advance the reader's understanding. The fundamental perspective
is that there is no best way to solve a given ordinary differential equation.
Indeed, most equations that scientists encounter are solved numerically, and
the traditional analytical techniques presented here remain important because
they provide the basis for successful computing schemes. Readers are definitely
urged to use available software to learn about the solutions of the differential
equations they either need to solve or have otherwise become fascinated by.
The examples and exercises included have been collected over many years
for various classes given by the author. Many were taken from others' textbooks
and papers (only a few are original), so it is no longer possible to properly ac-
knowledge the original sources. This explicit debt to earlier writers is, certainly,
substantial. Likewise, little reference is made to more advanced monographs,
as would be appropriate for students seeking a less-utilitarian acquintanceship
with differential equations.
We hope readers will find the problems considered interesting and challeng-
ing. Moreover, we hope they will learn enough about how to solve differential
IX
