Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

Cover......Page 1
Title Page......Page 7
Copyright......Page 8
Contents......Page 23
1.1 Mathematical Models, Solutions, and Direction Fields......Page 27
Terminology......Page 29
Solutions and Integral Curves......Page 30
The Direction Field for u' = –k(u – T)......Page 32
Solutions and Direction Fields for y'  = f (t, y)......Page 36
Constructing Mathematical Models......Page 39
1.2 Linear Equations: Method of Integrating Factors......Page 43
The Method of Integrating Factors for Solving y' + p(t)y = g(t)......Page 46
1.3 Numerical Approximations: Euler’s Method......Page 53
1.4 Classification of Differential Equations......Page 63
CHAPTER SUMMARY......Page 69
2 First Order Differential Equations......Page 70
2.1 Separable Equations......Page 71
2.2 Modeling with First Order Equations......Page 78
2.3 Differences Between Linear and Nonlinear Equations......Page 93
2.4 Autonomous Equations and Population Dynamics......Page 103
2.5 Exact Equations and Integrating Factors......Page 117
2.6 Accuracy of Numerical Methods......Page 125
2.7 Improved Euler and Runge–Kutta Methods......Page 133
CHAPTER SUMMARY......Page 140
2.P.1 Harvesting a Renewable Resource......Page 141
2.P.2 Designing a Drip Dispenser for a Hydrology Experiment......Page 142
2.P.3 A Mathematical Model of a Groundwater Contaminant Source......Page 144
2.P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin......Page 146
3 Systems of Two First Order Equations......Page 149
3.1 Systems of Two Linear Algebraic Equations......Page 150
Eigenvalues and Eigenvectors......Page 155
3.2 Systems of Two First Order Linear Differential Equations......Page 162
Matrix Notation, Vector Solutions, and Component Plots......Page 165
Geometry of Solutions: Direction Fields and Phase Portraits......Page 167
Phase Portraits......Page 169
General Solutions of Two First Order Linear Equations......Page 170
Linear Autonomous Systems......Page 171
Transformation of a Second Order Equation to a System of First Order Equations......Page 173
3.3 Homogeneous Linear Systems with Constant Coefficients......Page 179
On Reducing x'  = Ax + b to x'  = Ax......Page 180
The Eigenvalue Method for Solving x'  = Ax......Page 181
Extension to a General System......Page 182
Real and Different Eigenvalues......Page 183
Nodal Sources and Nodal Sinks......Page 188
Saddle Points......Page 191
3.4 Complex Eigenvalues......Page 195
Spiral Points and Centers......Page 200
3.5 Repeated Eigenvalues......Page 205
3.6 A Brief Introduction to Nonlinear Systems......Page 216
3.7 Numerical Methods for Systems of First Order Equations......Page 224
CHAPTER SUMMARY......Page 227
3.P.1 Eigenvalue-Placement Design of a Satellite Attitude Control System......Page 229
3.P.2 Estimating Rate Constants for an Open Two-Compartment Model......Page 232
3.P.3 The Ray Theory of Wave Propagation......Page 234
3.P.4 A Blood-Brain Pharmacokinetic Model......Page 237
4.1 Definitions and Examples......Page 240
Linear Equations......Page 241
Dynamical System Formulation......Page 242
The Spring-Mass System......Page 243
The Linearized Pendulum......Page 248
The Series RLC Circuit......Page 250
Existence and Uniqueness of Solutions......Page 253
Linear Operators and the Principle of Superposition for Linear Homogeneous Equations......Page 256
Wronskians and Fundamental Sets of Solutions......Page 258
Abel’s Equation for the Wronskian......Page 261
4.3 Linear Homogeneous Equations with Constant Coefficients......Page 265
The Characteristic Equation of ay" + by' + cy = 0......Page 266
Initial Value Problems and Phase Portraits......Page 270
4.4 Mechanical and Electrical Vibrations......Page 278
Undamped Free Vibrations......Page 279
Damped Free Vibrations......Page 282
Phase Portraits for Harmonic Oscillators......Page 286
4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients......Page 291
General Solution Strategy......Page 292
Method of Undetermined Coefficients......Page 293
Superposition Principle for Nonhomogeneous Equations......Page 295
Summary: Method of Undetermined Coefficients......Page 297
Forced Vibrations with Damping......Page 300
The Frequency Response Function......Page 302
Forced Vibrations without Damping......Page 307
4.7 Variation of Parameters......Page 311
Variation of Parameters for Linear Second Order Equations......Page 315
CHAPTER SUMMARY......Page 319
4.P.1 A Vibration Insulation Problem......Page 322
4.P.2 Linearization of a Nonlinear Mechanical System......Page 323
4.P.3 A Spring-Mass Event Problem......Page 325
4.P.4 Uniformly Distributing Points on a Sphere......Page 326
4.P.5 Euler–Lagrange Equations......Page 330
5 The Laplace Transform......Page 335
5.1 Definition of the Laplace Transform......Page 336
Existence of the Laplace Transform......Page 342
5.2 Properties of the Laplace Transform......Page 345
5.3 The Inverse Laplace Transform......Page 352
Partial Fractions......Page 356
5.4 Solving Differential Equations with Laplace Transforms......Page 361
Characteristic Polynomials and Laplace Transforms of Differential Equations......Page 364
5.5 Discontinuous Functions and Periodic Functions......Page 369
5.6 Differential Equations with Discontinuous Forcing Functions......Page 378
5.7 Impulse Functions......Page 385
5.8 Convolution Integrals and Their Applications......Page 392
5.9 Linear Systems and Feedback Control......Page 402
CHAPTER SUMMARY......Page 410
5.P.2 Effects of Pole Locations on Step Responses of Second Order Systems......Page 412
5.P.3 The Watt Governor, Feedback Control, and Stability......Page 415
6 Systems of First Order Linear Equations......Page 420
6.1 Definitions and Examples......Page 421
First Order Linear Systems: General Framework......Page 422
Applications Modeled by First Order Linear Systems......Page 423
6.2 Basic Theory of First Order Linear Systems......Page 432
6.3 Homogeneous Linear Systems with Constant Coefficients......Page 442
The Matrix A Is Nondefective With Real Eigenvalues......Page 443
6.4 Nondefective Matrices with Complex Eigenvalues......Page 453
Fundamental Matrices......Page 463
The Matrix Exponential Function e[sup(At)]......Page 466
Methods for Constructing e[sup(At)]......Page 469
Variation of Parameters......Page 474
Undetermined Coefficients and Frequency Response......Page 477
6.7 Defective Matrices......Page 481
CHAPTER SUMMARY......Page 487
6.P.1 A Compartment Model of Heat Flow in a Rod......Page 489
6.P.2 Earthquakes and Tall Buildings......Page 492
6.P.3 Controlling a Spring-Mass System to Equilibrium......Page 494
7.1 Autonomous Systems and Stability......Page 502
7.2 Almost Linear Systems......Page 512
7.3 Competing Species......Page 523
7.4 Predator–Prey Equations......Page 534
7.5 Periodic Solutions and Limit Cycles......Page 543
7.6 Chaos and Strange Attractors: The Lorenz Equations......Page 554
CHAPTER SUMMARY......Page 561
7.P.1 Modeling of Epidemics......Page 562
7.P.2 Harvesting in a Competitive Environment......Page 564
7.P.3 The Rossler System......Page 566
A.1 Matrices......Page 567
A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank......Page 576
A.3 Determinants and Inverses......Page 593
A.4 The Eigenvalue Problem......Page 602
B: Complex Variables......Page 613
ANSWERS TO SELECTED PROBLEMS......Page 618
REFERENCES......Page 704
PHOTO CREDITS......Page 705
INDEX......Page 706