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[资源] Differential Equations An Introduction to Modern Methods and Applications

C O N T E N T S
1 Introduction 1
1.1 Mathematical Models, Solutions, and Direction Fields 1
1.2 Linear Equations: Method of Integrating Factors 17
1.3 Numerical Approximations: Euler’s Method 27
1.4 Classification of Differential Equations 37
2 First Order Differential Equations 44
2.1 Separable Equations 45
2.2 Modeling with First Order Equations 52
2.3 Differences Between Linear and Nonlinear Equations 67
2.4 Autonomous Equations and Population Dynamics 77
2.5 Exact Equations and Integrating Factors 91
2.6 Accuracy of Numerical Methods 99
2.7 Improved Euler and Runge–Kutta Methods 107
Projects
2.P.1 Harvesting a Renewable Resource 115
2.P.2 Designing a Drip Dispenser for a Hydrology Experiment 116
2.P.3 A Mathematical Model of a Groundwater Contaminant Source 118
2.P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin 120
3 Systems of Two First Order Equations 123
3.1 Systems of Two Linear Algebraic Equations 124
3.2 Systems of Two First Order Linear Differential Equations 136
3.3 Homogeneous Linear Systems with Constant Coefficients 153
3.4 Complex Eigenvalues 169
3.5 Repeated Eigenvalues 179
3.6 A Brief Introduction to Nonlinear Systems 190
3.7 Numerical Methods for Systems of First Order Equations 198
Projects
3.P.1 Eigenvalue-Placement Design of a Satellite Attitude Control System 203
3.P.2 Estimating Rate Constants for an Open Two-Compartment Model 206
3.P.3 The Ray Theory of Wave Propagation 208
3.P.4 A Blood-Brain Pharmacokinetic Model 211
4 Second Order Linear Equations 214
4.1 Definitions and Examples 214
4.2 Theory of Second Order Linear Homogeneous Equations 227
4.3 Linear Homogeneous Equations with Constant Coefficients 239
4.4 Mechanical and Electrical Vibrations 252
4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 265
4.6 Forced Vibrations, Frequency Response, and Resonance 274
4.7 Variation of Parameters 285
xxi
xxii Contents
Projects
4.P.1 A Vibration Insulation Problem 296
4.P.2 Linearization of a Nonlinear Mechanical System 297
4.P.3 A Spring-Mass Event Problem 299
4.P.4 Uniformly Distributing Points on a Sphere 300
4.P.5 Euler–Lagrange Equations 304
5 The Laplace Transform 309
5.1 Definition of the Laplace Transform 310
5.2 Properties of the Laplace Transform 319
5.3 The Inverse Laplace Transform 326
5.4 Solving Differential Equations with Laplace Transforms 335
5.5 Discontinuous Functions and Periodic Functions 343
5.6 Differential Equations with Discontinuous Forcing Functions 352
5.7 Impulse Functions 359
5.8 Convolution Integrals and Their Applications 366
5.9 Linear Systems and Feedback Control 376
Projects
5.P.1 An Electric Circuit Problem 386
5.P.2 Effects of Pole Locations on Step Responses of Second Order Systems 386
5.P.3 The Watt Governor, Feedback Control, and Stability 389
6 Systems of First Order Linear Equations 394
6.1 Definitions and Examples 395
6.2 Basic Theory of First Order Linear Systems 406
6.3 Homogeneous Linear Systems with Constant Coefficients 416
6.4 Nondefective Matrices with Complex Eigenvalues 427
6.5 Fundamental Matrices and the Exponential of a Matrix 437
6.6 Nonhomogeneous Linear Systems 448
6.7 Defective Matrices 455
Projects
6.P.1 A Compartment Model of Heat Flow in a Rod 463
6.P.2 Earthquakes and Tall Buildings 466
6.P.3 Controlling a Spring-Mass System to Equilibrium 468
7 Nonlinear Differential Equations and Stability 476
7.1 Autonomous Systems and Stability 476
7.2 Almost Linear Systems 486
7.3 Competing Species 497
7.4 Predator–Prey Equations 508
7.5 Periodic Solutions and Limit Cycles 517
7.6 Chaos and Strange Attractors: The Lorenz Equations 528
Projects
7.P.1 Modeling of Epidemics 536
7.P.2 Harvesting in a Competitive Environment 538
7.P.3 The Ro¨ ssler System 540
Contents xxiii
A Matrices and Linear Algebra 541
A.1 Matrices 541
A.2 Systems of Linear Algebraic Equations, Linear Independence,
and Rank 550
A.3 Determinants and Inverses 567
A.4 The Eigenvalue Problem 576
B Complex Variables 587
ANSWERS TO SELECTED PROBLEMS 592
REFERENCES 678
PHOTO CREDITS 679
INDEX 680

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