A Textbook on Ordinary Differential Equations

337 Pages·2015·2.941 MB·English

by  Shair Ahmad, Antonio Ambrosetti

Shair Ahmad
Antonio Ambrosetti
A Textbook
on Ordinary T
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Differential
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Equations
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Second Edition
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UNITEXT – La Matematica per il 3+2
Volume 88
Moreinformationaboutthisseriesat
http://www.springer.com/series/5418
(cid:2)
Shair Ahmad Antonio Ambrosetti
A Textbook on Ordinary
Differential Equations
Second Edition
ShairAhmad AntonioAmbrosetti
DepartmentofMathematics SISSA
UniversityofTexasatSanAntonio Trieste,Italy
SanAntonio,USA
UNITEXT–LaMatematicaperil3+2
ISSN2038-5722 ISSN2038-5757(electronic)
ISBN978-3-319-16407-6 ISBN978-3-319-16408-3(ebook)
DOI10.1007/978-3-319-16408-3
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2015932461
©SpringerInternationalPublishingSwitzerland2015
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Professor Shair Ahmadwishes to thankhis
wife, Carol, for her continuedand loving
support, patienceandunderstanding, which
gofar beyondwhatmightnormally be
expected. He alsowishes to acknowledgehis
grandson, AltonShairson, asa source of
infusionofenergy andoptimisticenthusiasm
for life.
Preface
Oneoftheauthors’mainmotivationsforwritingthisbookhasbeentoprovidestu-
dentsandfacultywithamoreeconomicaloptionwhenselectinganintroductorytext-
bookonordinarydifferentialequations(ODEs).Thisbookisaprimeronthetheory
andapplicationsofODEs.ItisaimedatstudentsofMathematics,Physics,Engineer-
ing,Statistics,InformationScience,etc.whoalreadypossessasufficientknowledge
ofcalculusandaminimalknowledgeoflinearalgebra.
The first chapter starts with the simplest first-order linear differentialequations
andbuildsonthistoleadtothemoregeneralequations.Theconceptsofinitialvalues
andexistenceanduniquenessofsolutionsareintroducedearlyinthischapter.Ample
examples,usingsimpleintegration,areprovidedtomotivateanddemonstratethese
concepts.Almostalloftheassertionsareprovedinelementaryandsimpleterms.
TheimportantconceptsoftheCauchyproblemandtheexistenceanduniqueness
ofsolutionsare covered in detailand demonstrated by many examples. Proofs are
giveninanAppendix.Thereisalsoarigoroustreatmentofsomequalitativebehav-
iorofsolutions.Thischapterisimportantfromapedagogicalpointofviewbecause
itintroducesstudentstorigorandfostersanunderstandingofimportantconceptsat
anearlystage.
Inachapteronnonlinearfirst-orderequations,studentswilllearnhowtoexplicitly
solvecertain typesofequations, such as separable, homogeneous, exact, Bernoulli
andClairautequations.Furtherchaptersaredevotedtolinearhigherorderequations
andsystems,withseveralapplicationstomechanicsandelectricalcircuittheory.Also
includedisanelementarybutrigorousintroductiontothetheoryofoscillation.
Achapteronphaseplaneanalysisdealswithfindingperiodicsolutions,solutions
of simple boundary value problems, and homoclinic and heteroclinic trajectories.
There is also a sectionon the Lotka–Volterrasystem in thearea ofpopulationdy-
namics.
Subsequently,thebookdealswiththeSturm–Liouvilleeigenvalues,Laplacetrans-
formandfindingseriessolutions,includingfairlydetailedtreatmentofBesselfunc-
tions,whichareimportantinEngineering.
Althoughthisbookismainlyaddressedtoundergraduatestudents,consideration
isgiventosomemoreadvancedtopics,suchasstabilitytheoryandexistenceofso-
viii Preface
lutionstoboundaryvalueproblems,whichmightbeusefulformoremotivatedun-
dergraduatesorevenbeginninggraduatestudents.
A chapter on numerical methods is included as an Appendix,where the impor-
tanceofcomputertechnologyispointedout.Otherwise,wedonotencouragetheuse
ofcomputertechnologyatthislevel.Webelievethat,atthisstage, studentsshould
practicetheirpriorknowledgeofalgebraandcalculusinsteadofrelyingontechnol-
ogy,thussharpeningtheirmathematicalskillsingeneral.
Each chapter ends witha set of exercises that are intended to test the student’s
understandingoftheconceptscovered. Solutionstoselected exercises areincluded
attheendofthebook.
WewishtoacknowledgewithgratitudethehelpofDungLe,RahbarMaghsoudi
andVittorioCotiZelati,especiallywithregardtotechnicalissues.
SanAntonioandTrieste ShairAhmad
December2013 AntonioAmbrosetti
Preface to the Second Edition
Thiseditioncontainscorrectionsoferrataandadditionalcarefullyselectedexercises
andprovidesmorelucidexplanationsofsomeofthetopicsaddressed.Althoughthe
bookiswrittenin a rigorousand thoroughstyle,it offers instructorsthe flexibility
toskipsome of the rigorand theoryand concentrate on methodsand applications,
shouldtheywishtodoso.Thismakesthebooksuitablenotonlyforstudentsstudy-
ingMathematics butalso for those inother areas ofScience and Engineering. We
wishtothankWeimingCaoandErikWhalénforseveralusefulcomments.
SanAntonioandTrieste ShairAhmad
January2015 AntonioAmbrosetti
Contents
1 Firstorderlineardifferentialequations........................... 1
1.1 Introduction ............................................... 1
1.2 Asimplecase.............................................. 2
1.3 Someexamplesarisinginapplications......................... 3
1.3.1 Populationdynamics ................................. 3
1.3.2 AnRCelectriccircuit ................................ 4
1.4 Thegeneralcase ........................................... 5
1.5 Exercises ................................................. 13
2 Theoryoffirstorderdifferentialequations........................ 15
2.1 Differentialequationsandtheirsolutions....................... 15
2.2 TheCauchyproblem:Existenceanduniqueness................. 17
2.2.1 Localexistenceanduniqueness ........................ 18
2.2.2 Globalexistenceanduniqueness ....................... 24
2.3 Qualitativepropertiesofsolutions............................. 25
2.4 Improvingtheexistenceanduniquenessresults ................. 30
2.5 Appendix:Proofofexistenceanduniquenesstheorems........... 32
2.5.1 ProofofTheorem2.4.5 ............................... 32
2.5.2 ProofofTheorem2.4.4 ............................... 36
2.6 Exercises ................................................. 37
3 Firstordernonlineardifferentialequations ....................... 39
3.1 Separableequations......................................... 39
3.1.1 Thelogisticequation ................................. 41
3.2 Exactequations ............................................ 44
3.3 Theintegratingfactor ....................................... 54
3.4 Homogeneousequations..................................... 58
3.5 Bernoulliequations......................................... 61
3.6 Appendix.SingularsolutionsandClairautequations ............. 62
3.6.1 Clairautequations.................................... 64
3.7 Exercises ................................................. 67
x Contents
4 Existenceanduniquenessforsystemsandhigherorderequations ... 71
4.1 Systemsofdifferentialequations.............................. 71
4.1.1 Existenceanduniquenessresultsforsystems ............. 73
4.2 Higherorderequations ...................................... 74
4.2.1 Existenceanduniquenessforn-thorderequations......... 75
4.3 Exercises ................................................. 76
5 Secondorderequations ......................................... 79
5.1 Linearhomogeneousequations ............................... 79
5.2 LinearindependenceandtheWronskian ....................... 83
5.2.1 Wronskian.......................................... 85
5.3 Reductionoftheorder ...................................... 88
5.4 Linearnonhomogeneousequations............................ 89
5.4.1 Variationofparameters ............................... 91
5.5 Linearhomogeneousequationswithconstantcoefficients......... 93
5.5.1 TheEulerequation................................... 100
5.6 Linearnonhomogeneousequations–methodofundetermined
coefficients................................................. 101
5.6.1 Theelasticspring .................................... 108
5.7 Oscillatorybehaviorofsolutions.............................. 110
5.8 Somenonlinearsecondorderequations ........................ 116
5.8.1 EquationsofthetypeF.t;x0;x00/D0 .................. 116
5.8.2 EquationsofthetypeF.x;x0;x00/D0 ................. 116
5.8.3 EquationsoftheformF.t;x;x0;x00/D0withF homogenous117
5.9 Exercises ................................................. 119
6 Higherorderlinearequations ................................... 125
6.1 Existenceanduniqueness.................................... 125
6.2 LinearindependenceandWronskian .......................... 126
6.3 Constantcoefficients........................................ 127
6.4 Nonhomogeneousequations ................................. 130
6.5 Exercises ................................................. 133
7 Systemsoffirstorderequations.................................. 135
7.1 Preliminaries:Abriefreviewoflinearalgebra .................. 135
7.1.1 Basicpropertiesofmatrices ........................... 135
7.1.2 Determinants........................................ 136
7.1.3 Inverseofamatrix ................................... 139
7.1.4 Eigenvaluesandeigenvectors.......................... 140
7.1.5 TheJordannormalform .............................. 142
7.2 Firstordersystems ......................................... 144
7.3 Linearfirstordersystems .................................... 146
7.3.1 Wronskianandlinearindependence..................... 148
7.4 Constantsystems–eigenvaluesandeigenvectors................ 152
7.5 Nonhomogeneoussystems ................................... 158
7.6 Exercises ................................................. 162