An introduction to ordinary differential equations

415 Pages·2004·3.632 MB·English

by  James C. Robinson

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AN INTRODUCTION TO ORDINARY
DIFFERENTIAL EQUATIONS
Thisrefreshing,introductorytextbookcoversstandardtechniquesforsolvingordi-
nary differential equations, as well as introducing students to qualitative methods
suchasphase-planeanalysis.Thepresentationisconcise,informalyetrigorous;it
canbeusedforeitherone-termorone-semestercourses.
TopicssuchasEuler’smethod,differenceequations,thedynamicsofthelogistic
mapandtheLorenzequations,demonstratethevitalityofthesubject,andprovide
pointers to further study. The author also encourages a graphical approach to the
equationsandtheirsolutions,andtothatendthebookisprofuselyillustrated.The
MATLABfilesusedtoproducemanyofthefiguresareprovidedinanaccompany-
ingwebsite.
Numerous worked examples provide motivation for, and illustration of, key
ideas and show how to make the transition from theory to practice. Exercises are
alsoprovidedtotestandextendunderstanding;fullsolutionsfortheseareavailable
forteachers.

AN INTRODUCTION TO ORDINARY
DIFFERENTIAL EQUATIONS
JAMES C. ROBINSON
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
 
The Edinburgh Building, Cambridge , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
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© Cambridge University Press 2004
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without the written permission of Cambridge University Press.
First published in print format 2004
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Contents
Preface page xiii
Introduction 1
PartI Firstorderdifferentialequations 3
1 Radioactivedecayandcarbondating 5
1.1 Radioactivedecay 5
1.2 Radiocarbondating 6
Exercises 8
2 Integrationvariables 9
3 Classificationofdifferentialequations 11
3.1 Ordinaryandpartialdifferentialequations 11
3.2 Theorderofadifferentialequation 13
3.3 Linearandnonlinear 13
3.4 Differenttypesofsolution 14
Exercises 16
4 *Graphicalrepresentationofsolutions
usingMATLAB 18
Exercises 21
5 ‘Trivial’differentialequations 22
5.1 TheFundamentalTheoremofCalculus 22
5.2 Generalsolutionsandinitialconditions 25
5.3 Velocity,accelerationandNewton’ssecondlaw
ofmotion 29
5.4 Anequationthatwecannotsolveexplicitly 32
Exercises 33
Someofthechapters,andsomesectionswithinotherchapters,aremarkedwithanasterisk(*).Thesepartsofthe
bookcontainmaterialthateitherismoreadvanced,orexpandsonpointsraisedelsewhereinthetext.
vii
viii Contents
6 Existenceanduniquenessofsolutions 38
6.1 Thecaseforanabstractresult 38
6.2 Theexistenceanduniquenesstheorem 40
6.3 Maximalintervalofexistence 41
6.4 TheClayMathematicsInstitute’s$1000000
question 42
Exercises 44
7 ScalarautonomousODEs 46
7.1 Thequalitativeapproach 46
7.2 Stability,instabilityandbifurcation 48
7.3 Analyticconditionsforstabilityandinstability 49
7.4 Structuralstabilityandbifurcations 50
7.5 Someexamples 50
7.6 Thepitchforkbifurcation 54
7.7 Dynamicalsystems 56
Exercises 56
8 Separableequations 59
8.1 Thesolution‘recipe’ 59
8.2 Thelinearequation x˙ = λx 61
8.3 Malthus’populationmodel 62
8.4 Justifyingthemethod 64
8.5 Amorerealisticpopulationmodel 66
8.6 Furtherexamples 68
Exercises 72
9 Firstorderlinearequationsandtheintegratingfactor 75
9.1 Constantcoefficients 75
9.2 Integratingfactors 76
9.3 Examples 78
9.4 Newton’slawofcooling 79
Exercises 86
10 Two‘tricks’fornonlinearequations 89
10.1 Exactequations 89
10.2 Substitutionmethods 94
Exercises 97
PartII Secondorderlinearequationswithconstantcoefficients 99
11 Secondorderlinearequations:generaltheory 101
11.1 Existenceanduniqueness 101
11.2 Linearity 102
11.3 Linearlyindependentsolutions 104
11.4 *TheWronskian 106