Differential Equations with Applications and Historical Notes, Third Edition

763 Pages·2016·11.72 MB·English

by  George F. Simmons

DIFFERENTIAL EQUATIONS
WITH APPLICATIONS AND
HISTORICAL NOTES
Third Edition
TEXTBOOKS in MATHEMATICS
Series Editors: Al Boggess and Ken Rosen
PUBLISHED TITLES
ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION
William Paulsen
ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH
Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom
ADVANCED LINEAR ALGEBRA
Hugo Woerdeman
APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION
Richard Klima, Neil Sigmon, and Ernest Stitzinger
APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE
Vladimir Dobrushkin
COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI,
SECOND EDITION
Robert E. White
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION
Steven G. Krantz
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS
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
EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala
GRAPHS & DIGRAPHS, SIXTH EDITION
Gary Chartrand, Linda Lesniak, and Ping Zhang
INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION
Jonathan D. H. Smith
TEXTBOOKS in MATHEMATICS TEXTBOOKS in MATHEMATICS
Series Editors: Al Boggess and Ken Rosen
PUBLISHED TITLES
ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION DIFFERENTIAL EQUATIONS
William Paulsen
ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH WITH APPLICATIONS AND
Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom
HISTORICAL NOTES
ADVANCED LINEAR ALGEBRA
Hugo Woerdeman
Third Edition
APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION
Richard Klima, Neil Sigmon, and Ernest Stitzinger
APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE
Vladimir Dobrushkin
COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI,
SECOND EDITION
Robert E. White
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION
Steven G. Krantz
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS
Steven G. Krantz George F. Simmons
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
EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala
GRAPHS & DIGRAPHS, SIXTH EDITION
Gary Chartrand, Linda Lesniak, and Ping Zhang
INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION
Jonathan D. H. Smith
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Contents
Preface to the Third Edition .................................................................................xi
Preface to the Second Edition ............................................................................xiii
Preface to the First Edition ...................................................................................xv
Suggestions for the Instructor ...........................................................................xix
About the Author ................................................................................................xxi
1. The Nature of Differential Equations. Separable Equations ................1
1 Introduction ...........................................................................................1
2 General Remarks on Solutions ............................................................4
3 Families of Curves. Orthogonal Trajectories ..................................11
4 Growth, Decay, Chemical Reactions, and Mixing..........................19
5 Falling Bodies and Other Motion Problems ...................................31
6 The Brachistochrone. Fermat and the Bernoullis ...........................40
Appendix A: Some Ideas From the Theory of Probability:
The Normal Distribution Curve (or Bell Curve) and
Its Differential Equation ................................................................................51
2. First Order Equations ..................................................................................65
7 Homogeneous Equations ...................................................................65
8 Exact Equations ...................................................................................69
9 Integrating Factors ..............................................................................74
10 Linear Equations .................................................................................81
11 Reduction of Order .............................................................................85
12 The Hanging Chain. Pursuit Curves ...............................................88
13 Simple Electric Circuits ......................................................................95
3. Second Order Linear Equations...............................................................107
14 Introduction .......................................................................................107
15 The General Solution of the Homogeneous Equation .................113
16 The Use of a Known Solution to find Another .............................119
17 The Homogeneous Equation with Constant Coefficients ...........122
18 The Method of Undetermined Coefficients ..................................127
19 The Method of Variation of Parameters .........................................133
20 Vibrations in Mechanical and Electrical Systems ........................136
21 Newton’s Law of Gravitation and The Motion of the Planets ......146
22 Higher Order Linear Equations. Coupled
Harmonic Oscillators .......................................................................155
23 Operator Methods for Finding Particular Solutions ....................161
Appendix A. Euler .......................................................................................170
Appendix B. Newton ...................................................................................179
vii
viii Contents
4. Qualitative Properties of Solutions ........................................................187
24 Oscillations and the Sturm Separation Theorem .........................187
25 The Sturm Comparison Theorem ...................................................194
5. Power Series Solutions and Special Functions .....................................197
26 Introduction. A Review of Power Series ........................................197
27 Series Solutions of First Order Equations ......................................206
28 Second Order Linear Equations. Ordinary Points .......................210
29 Regular Singular Points ...................................................................219
30 Regular Singular Points (Continued) .............................................229
31 Gauss’s Hypergeometric Equation .................................................236
32 The Point at Infinity ..........................................................................242
Appendix A. Two Convergence Proofs ....................................................246
Appendix B. Hermite Polynomials and Quantum Mechanics ..............250
Appendix C. Gauss ......................................................................................262
Appendix D. Chebyshev Polynomials and the Minimax Property ......270
Appendix E. Riemann’s Equation .............................................................278
6. Fourier Series and Orthogonal Functions .............................................289
33 The Fourier Coefficients ...................................................................289
34 The Problem of Convergence ..........................................................301
35 Even and Odd Functions. Cosine and Sine Series .......................310
36 Extension to Arbitrary Intervals .....................................................319
37 Orthogonal Functions ......................................................................325
38 The Mean Convergence of Fourier Series ......................................336
Appendix A. A Pointwise Convergence Theorem ..................................345
7. Partial Differential Equations and Boundary Value Problems ........351
39 Introduction. Historical Remarks ...................................................351
40 Eigenvalues, Eigenfunctions, and the Vibrating String ..............355
41 The Heat Equation ............................................................................366
42 The Dirichlet Problem for a Circle. Poisson’s Integral .................372
43 Sturm–Liouville Problems ...............................................................379
Appendix A. The Existence of Eigenvalues and Eigenfunctions ..........388
8. Some Special Functions of Mathematical Physics ..............................393
44 Legendre Polynomials ......................................................................393
45 Properties of Legendre Polynomials ..............................................400
46 Bessel Functions. The Gamma Function .......................................407
47 Properties of Bessel Functions ........................................................418
Appendix A. Legendre Polynomials and Potential Theory ...................427
Appendix B. Bessel Functions and the Vibrating Membrane ................435
Appendix C. Additional Properties of Bessel Functions ........................441
Contents ix
9. Laplace Transforms ....................................................................................447
48 Introduction .......................................................................................447
49 A Few Remarks on the Theory .......................................................452
50 Applications to Differential Equations ..........................................457
51 Derivatives and Integrals of Laplace Transforms ........................463
52 Convolutions and Abel’s Mechanical Problem .............................468
53 More about Convolutions. The Unit Step and
Impulse Functions ............................................................................475
Appendix A. Laplace ...................................................................................483
Appendix B. Abel .........................................................................................484
10. Systems of First Order Equations............................................................487
54 General Remarks on Systems ..........................................................487
55 Linear Systems ...................................................................................491
56 Homogeneous Linear Systems with Constant Coefficients ........498
57 Nonlinear Systems. Volterra’s Prey-Predator Equations .............507
11. Nonlinear Equations ..................................................................................513
58 Autonomous Systems. The Phase Plane and Its Phenomena .....513
59 Types of Critical Points. Stability ....................................................519
60 Critical Points and Stability for Linear Systems ...........................529
61 Stability By Liapunov’s Direct Method ..........................................541
62 Simple Critical Points of Nonlinear Systems ................................547
63 Nonlinear Mechanics. Conservative Systems ...............................557
64 Periodic Solutions. The Poincaré–Bendixson Theorem ...............563
65 More about the van der Pol Equation.............................................572
Appendix A. Poincaré .................................................................................574
Appendix B. Proof of Liénard’s Theorem ................................................576
12. The Calculus of Variations .......................................................................581
66 Introduction. Some Typical Problems of the Subject ...................581
67 Euler’s Differential Equation for an Extremal ..............................584
68 Isoperimetric Problems ....................................................................595
Appendix A. Lagrange ................................................................................606
Appendix B. Hamilton’s Principle and Its Implications ........................608
13. The Existence and Uniqueness of Solutions .........................................621
69 The Method of Successive Approximations ..................................621
70 Picard’s Theorem ...............................................................................626
71 Systems. The Second Order Linear Equation ...............................638
14. Numerical Methods ...................................................................................643
By John S. Robertson
72 Introduction .......................................................................................643