Applied differential equations: an introduction

723 Pages·2015·18.24 MB·English

by  Dobrushkin, Vladimir Andreevich

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TEXTBOOKS in MATHEMATICS
APPLIED
DIFFERENTIAL
EQUATIONS
The Primary Course
Vladimir A. Dobrushkin
Brown University
Providence, Rhode Island, USA
K12330_FM.indd 1 11/17/14 11:50 AM
TEXTBOOKS in MATHEMATICS
Series Editors: Al Boggess and Ken Rosen
PUBLISHED TITLES
ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH
Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom
ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH
William Paulsen
ADVANCED CALCULUS: THEORY AND PRACTICE
John Srdjan Petrovic
ADVANCED LINEAR ALGEBRA
Nicholas Loehr
ANALYSIS WITH ULTRASMALL NUMBERS
Karel Hrbacek, Olivier Lessmann, and Richard O’Donovan
APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE
Vladimir Dobrushkin
APPLYING ANALYTICS: A PRACTICAL APPROACH
Evan S. Levine
COMPUTATIONS OF IMPROPER REIMANN INTEGRALS
Ioannis Roussos
CONVEX ANALYSIS
Steven G. Krantz
COUNTEREXAMPLES: FROM ELEMENTARY CALCULUS TO THE BEGINNINGS OF ANALYSIS
Andrei Bourchtein and Ludmila Bourchtein
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION
Steven G. Krantz
DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY
Mark A. McKibben and Micah D. Webster
ELEMENTARY NUMBER THEORY
James S. Kraft and Lawrence C. Washington
ELEMENTS OF ADVANCED MATHEMATICS, THIRD EDITION
Steven G. Krantz
K12330_FM.indd 2 11/17/14 11:50 AM
PUBLISHED TITLES CONTINUED
EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala
AN INTRODUCTION TO NUMBER THEORY WITH CRYPTOGRAPHY
James Kraft and Larry Washington
AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH MATLAB®, SECOND EDITION
Mathew Coleman
INTRODUCTION TO THE CALCULUS OF VARIATIONS AND CONTROL WITH MODERN APPLICATIONS
John T. Burns
INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS, SECOND EDITION
Charles E. Roberts, Jr.
LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION
Bruce Solomon
THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY
David G. Taylor
QUADRACTIC IRRATIONALS: AN INTRODUCTION TO CLASSICAL NUMBER THEORY
Franz Holter-Koch
REAL ANALYSIS AND FOUNDATIONS, THIRD EDITION
Steven G. Krantz
RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION
Bilal M. Ayyub
RISK MANAGEMENT AND SIMULATION
Aparna Gupta
TRANSFORMATIONAL PLANE GEOMETRY
Ronald N. Umble and Zhigang Han
K12330_FM.indd 3 11/17/14 11:50 AM
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Contents
List of Symbols xi
Preface xiii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Classification of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Solutions to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Particular and Singular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Review Questions for Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 First Order Equations 45
2.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.1 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2 Equations Reducible to Separable Equations. . . . . . . . . . . . . . . . . . . . 61
2.2.1 Equations with Homogeneous Coefficients . . . . . . . . . . . . . . . . . 62
2.2.2 Equations with Homogeneous Fractions . . . . . . . . . . . . . . . . . . 66
2.2.3 Equations with Linear Coefficients . . . . . . . . . . . . . . . . . . . . . 73
2.3 Exact Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.4 Simple Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.5 First Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . 100
2.6 Special Classes of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.6.1 The Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.6.2 The Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.6.3 Equations with the Dependent or Independent Variable Missing. . . . . 121
2.6.4 EquationsHomogeneouswithRespecttoTheirDependentVariableand
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.6.5 Equations Solvable for a Variable . . . . . . . . . . . . . . . . . . . . . . 126
v
vi Contents
2.7 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.7.1 Bifurcation Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.7.2 Validity Intervals of Autonomous Equations . . . . . . . . . . . . . . . . 139
Summary for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Review Questions for Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3 Numerical Methods 155
3.1 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.2 Euler’s Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.3 The Polynomial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
3.4 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3.5 The Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Summary for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Review Questions for Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4 Second and Higher Order Linear Differential Equations 211
4.1 Second and Higher Order Differential Equations. . . . . . . . . . . . . . . . . . 212
4.1.1 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
4.1.2 Exact Equations and Integrating Factors . . . . . . . . . . . . . . . . . 216
4.1.3 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.2 Linear Independence and Wronskians. . . . . . . . . . . . . . . . . . . . . . . . 223
4.3 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 230
4.4 Equations with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 234
4.5 Complex Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.6 Repeated Roots. Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . 245
4.6.1 Reduction of Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
4.7 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
4.7.1 The Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . 255
4.8 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
4.9 Bessel Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.9.1 Parametric Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . 278
4.9.2 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . 279
4.9.3 Related Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 280
Summary for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Review Questions for Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5 Laplace Transforms 301
5.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
5.2 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . 316
Contents vii
5.3 Discontinuous and Impulse Functions. . . . . . . . . . . . . . . . . . . . . . . . 326
5.4 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
5.4.1 Partial Fraction Decomposition . . . . . . . . . . . . . . . . . . . . . . . 339
5.4.2 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
5.4.3 The Residue Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
5.5 Homogeneous Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 351
5.5.1 Equations with Variable Coefficients . . . . . . . . . . . . . . . . . . . . 355
5.6 Nonhomogeneous Differential Equations . . . . . . . . . . . . . . . . . . . . . . 358
5.6.1 Differential Equations with Intermittent Forcing Functions . . . . . . . 362
Summary for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Review Questions for Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
6 Introduction to Systems of ODEs 381
6.1 Some ODE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
6.1.1 RLC-Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
6.1.2 Spring-Mass Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
6.1.3 The Euler–LagrangeEquation . . . . . . . . . . . . . . . . . . . . . . . 385
6.1.4 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
6.1.5 Laminated Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
6.1.6 Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
6.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
6.3 Linear Systems of First Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . 406
6.4 Reduction to a Single ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
6.5 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Summary for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Review Questions for Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
7 Topics from Linear Algebra 423
7.1 The Calculus of Matrix Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 423
7.2 Inverses and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
7.2.1 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 431
7.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
7.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
7.5 Sylvester’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
7.6 The Resolvent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
7.7 The Spectral Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . 461
Summary for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Review Questions for Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
viii Contents
8 Systems of Linear Differential Equations 481
8.1 Systems of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
8.1.1 The Euler Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . 489
8.2 Constant Coefficient Homogeneous Systems . . . . . . . . . . . . . . . . . . . . 491
8.2.1 Simple Real Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
8.2.2 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
8.2.3 Repeated Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
8.2.4 Qualitative Analysis of Linear Systems . . . . . . . . . . . . . . . . . . . 504
8.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
8.3.1 Equations with Constant Coefficients. . . . . . . . . . . . . . . . . . . . 511
8.4 Method of Undetermined Coefficients. . . . . . . . . . . . . . . . . . . . . . . . 517
8.5 The Laplace Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
8.6 Second Order Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Summary for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Review Questions for Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
9 Qualitative Theory of Differential Equations 539
9.1 Autonomous Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
9.1.1 Two-Dimensional Autonomous Equations . . . . . . . . . . . . . . . . . 542
9.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
9.2.1 Two-Dimensional Autonomous Equations . . . . . . . . . . . . . . . . . 551
9.2.2 Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
9.3 Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
9.3.1 Competing Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
9.3.2 Predator-PreyEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 563
9.3.3 Other Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . 568
9.4 Conservative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
9.4.1 Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
9.5 Lyapunov’s Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
9.6 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
9.6.1 Equations with Periodic Coefficients . . . . . . . . . . . . . . . . . . . . 596
Summary for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
Review Questions for Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
10 Orthogonal Expansions 609
10.1 Sturm–Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
10.2 Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
10.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
10.3.1 Music as Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625