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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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Time-Dependent Outside Temperature 46
Air Conditioning a Room 47
The Case of the Melting Snowman 49
Explorations 51
2 Introduction to ODEs
25
3 Some Cool ODEs
43
CONTENTS
xv
4 Second-Order Linear Equations 57
William Boyce & William Siegmann, Rensselaer Polytechnic Institute
Second-Order ODEs and the Architect 58
Undamped Oscillations 58
The Effect of Damping 60
Forced Oscillations 61
Beats 63
Electrical Oscillations: An Analogy 64 Seismographs 64 Explorations 69
5 Models of Motion 77
Robert Borrelli & Courtney Coleman, Harvey Mudd College Vectors 78
Forces and Newton's Laws 79 Dunk Tank 80
Longer to Rise or to Fall? 81 Indiana Newton 82 Ski Jumping 84 Explorations 85
6 First-Order Linear Systems 93
William Boyce & William Siegmann, Rensselaer Polytechnic Institute Background 94
Examples of Systems: Pizza and Video, Coupled Springs 94 Linear Systems with Constant Coefficients 95 Solution Formulas: Eigenvalues and Eigenvectors 97 Calculating Eigenvalues and Eigenvectors 98 Phase Portraits 99
Using ODE Architect to Find Eigenvalues and Eigenvectors 102 Separatrices 103 Parameter Movies 103 Explorations 105
7 Nonlinear Systems 115
Michael Branton, Stetson University
Linear vs. Nonlinear 116
The Geometry of Nonlinear Systems 116
Linearization 117
Separatrices and Saddle Points 120
Behavior of Solutions Away from Equilibrium Points 121 Bifurcation to a Limit Cycle 122 Higher Dimensions 123
Spinning Bodies: Stability of Steady Rotations 123 The Planar Double Pendulum 126 Explorations 129
8 Compartment Models
Courtney Coleman & Michael Moody, Harvey Mudd College
Lake Pollution 136 Allergy Relief 137 Lead in the Body 139 Equilibrium 141
The Autocatalator and a Hopf Bifurcation 142 Explorations 147
9 Population Models
Michael Moody, Harvey Mudd College
Modeling PopulationGrowth 156 The LogisticModel 156 Two-Species Population Models 158 Predator and Prey 159 Species Competition 160
Mathematical Epidemiology: The SIRModel 161 Explorations 163
10 The Pendulum and Its Friends
John Hubbard & Beverly West, Cornell University
Modeling Pendulum Motion 174 Conservative Systems: Integrals of Motion 176 The Effect of Damping 177 Separatrices 180 Pumping a Swing 182
Writing the Equations of Motion for Pumping a Swing 182 Geodesics 185
Geodesics on a Surface of Revolution 186 Geodesics on a Torus 188 Explorations 193
CONTENTS
135
155
173
CONTENTS
xvii
11 Applications of Series Solutions 203
Anne Noonburg & Ben Pollina, University of Hartford
Infinite Series 204 Recurrence Formulas 204 Ordinary Points 206 Regular Singular Points 207 Bessel Functions 209
Transforming Bessel's Equation to the Aging Spring Equation 210 Explorations 213
12 Chaos and Control 221
John Hubbard & Beverly West, Cornell University Introduction 222
Solutions as Functions of Time 222
Poincare Sections 223
Periodic Points 224
The Unforced Pendulum 224
The Damped Forced Pendulum 226
Tangled Basins, the Wada Property 226
Gaining Control 228
Explorations 231
13 Discrete Dynamical Systems 233
Thomas LoFaro, Washington State University
Equilibrium States 235 Linear versus Nonlinear Dynamics 236 Stability of a Discrete Dynamical System 237 Bifurcations 238
Periodic and Chaotic Dynamics 240
What is Chaos? 241
Complex Numbers and Functions 242
Iterating a Complex Function 243
Julia Sets, the Mandelbrot Set, and Cantor Dust 244
Explorations 249
Glossary
257
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Modeling with the ODE Architect
t
Pacific sardine population and harvest.
In the two decades from 1932 to 1951, the Pacific sardine fishery completely collapsed. In this chapter you will learn to use the ODE Architect to construct a mathematical model which describes this event rather well. This will have two purposes: it will familiarize you with the menus and features ofthe ODE Architect, and it will acquaint you with the principles of mathematical modeling.
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