Differential Equations with Applications and Historical Notes 3rd Edition by George F. Simmons, ISBN-13: 978-1032477145

Differential Equations with Applications and Historical Notes 3rd Edition by George F. Simmons, ISBN-13: 978-1032477145

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• Publisher: ‎ Chapman and Hall/CRC; 3rd edition (January 21, 2023)
• Language: ‎ English
• 762 pages
• ISBN-10: ‎ 1032477148
• ISBN-13: ‎ 978-1032477145

Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one’s own time. An unfortunate effect of the predominance of fads is that if a student doesn’t learn about such worthwhile topics as the wave equation, Gauss’s hypergeometric function, the gamma function, and the basic problems of the calculus of variations―among others―as an undergraduate, then he/she is unlikely to do so later.

The natural place for an informal acquaintance with such ideas is a leisurely introductory course on differential equations. Specially designed for just such a course, Differential Equations with Applications and Historical Notes takes great pleasure in the journey into the world of differential equations and their wide range of applications. The author―a highly respected educator―advocates a careful approach, using explicit explanation to ensure students fully comprehend the subject matter.

With an emphasis on modeling and applications, the long-awaited Third Edition of this classic textbook presents a substantial new section on Gauss’s bell curve and improves coverage of Fourier analysis, numerical methods, and linear algebra. Relating the development of mathematics to human activity―i.e., identifying why and how mathematics is used―the text includes a wealth of unique examples and exercises, as well as the author’s distinctive historical notes, throughout.

• Provides an ideal text for a one- or two-semester introductory course on differential equations
• Emphasizes modeling and applications
• Presents a substantial new section on Gauss’s bell curve
• Improves coverage of Fourier analysis, numerical methods, and linear algebra
• Relates the development of mathematics to human activity―i.e., identifying why and how mathematics is used

Table of Contents:

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Suggestions for the Instructor
About the Author
1. The Nature of Differential Equations. Separable Equations
1 Introduction
2 General Remarks on Solutions
3 Families of Curves. Orthogonal Trajectories
4 Growth, Decay, Chemical Reactions, and Mixing
5 Falling Bodies and Other Motion Problems
6 The Brachistochrone. Fermat and the Bernoullis
Appendix A: Some Ideas From the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation
2. First Order Equations
7 Homogeneous Equations
8 Exact Equations
9 Integrating Factors
10 Linear Equations
11 Reduction of Order
12 The Hanging Chain. Pursuit Curves
13 Simple Electric Circuits
3. Second Order Linear Equations
14 Introduction
15 The General Solution of the Homogeneous Equation
16 The Use of a Known Solution to find Another
17 The Homogeneous Equation with Constant Coefficients
18 The Method of Undetermined Coefficients
19 The Method of Variation of Parameters
20 Vibrations in Mechanical and Electrical Systems
21 Newton’s Law of Gravitation and The Motion of the Planets
22 Higher Order Linear Equations. Coupled Harmonic Oscillators
23 Operator Methods for Finding Particular Solutions
Appendix A. Euler
Appendix B. Newton
4. Qualitative Properties of Solutions
24 Oscillations and the Sturm Separation Theorem
25 The Sturm Comparison Theorem
5. Power Series Solutions and Special Functions
26 Introduction. A Review of Power Series
27 Series Solutions of First Order Equations
28 Second Order Linear Equations. Ordinary Points
29 Regular Singular Points
30 Regular Singular Points (Continued)
31 Gauss’s Hypergeometric Equation
32 The Point at Infinity
Appendix A. Two Convergence Proofs
Appendix B. Hermite Polynomials and Quantum Mechanics
Appendix C. Gauss
Appendix D. Chebyshev Polynomials and the Minimax Property
Appendix E. Riemann’s Equation
6. Fourier Series and Orthogonal Functions
33 The Fourier Coefficients
34 The Problem of Convergence
35 Even and Odd Functions. Cosine and Sine Series
36 Extension to Arbitrary Intervals
37 Orthogonal Functions
38 The Mean Convergence of Fourier Series
Appendix A. A Pointwise Convergence Theorem
7. Partial Differential Equations and Boundary Value Problems
39 Introduction. Historical Remarks
40 Eigenvalues, Eigenfunctions, and the Vibrating String
41 The Heat Equation
42 The Dirichlet Problem for a Circle. Poisson’s Integral
43 Sturm–Liouville Problems
Appendix A. The Existence of Eigenvalues and Eigenfunctions
8. Some Special Functions of Mathematical Physics
44 Legendre Polynomials
45 Properties of Legendre Polynomials
46 Bessel Functions. The Gamma Function
47 Properties of Bessel Functions
Appendix A. Legendre Polynomials and Potential Theory
Appendix B. Bessel Functions and the Vibrating Membrane
Appendix C. Additional Properties of Bessel Functions
9. Laplace Transforms
48 Introduction
49 A Few Remarks on the Theory
50 Applications to Differential Equations
51 Derivatives and Integrals of Laplace Transforms
52 Convolutions and Abel’s Mechanical Problem
53 More about Convolutions. The Unit Step and Impulse Functions
Appendix A. Laplace
Appendix B. Abel
10. Systems of First Order Equations
54 General Remarks on Systems
55 Linear Systems
56 Homogeneous Linear Systems with Constant Coefficients
57 Nonlinear Systems. Volterra’s Prey-Predator Equations
11. Nonlinear Equations
58 Autonomous Systems. The Phase Plane and Its Phenomena
59 Types of Critical Points. Stability
60 Critical Points and Stability for Linear Systems
61 Stability By Liapunov’s Direct Method
62 Simple Critical Points of Nonlinear Systems
63 Nonlinear Mechanics. Conservative Systems
64 Periodic Solutions. The Poincaré–Bendixson Theorem
65 More about the van der Pol Equation
Appendix A. Poincaré
Appendix B. Proof of Liénard’s Theorem
12. The Calculus of Variations
66 Introduction. Some Typical Problems of the Subject
67 Euler’s Differential Equation for an Extremal
68 Isoperimetric Problems
Appendix A. Lagrange
Appendix B. Hamilton’s Principle and Its Implications
13. The Existence and Uniqueness of Solutions
69 The Method of Successive Approximations
70 Picard’s Theorem
71 Systems. The Second Order Linear Equation
14. Numerical Methods
72 Introduction
73 The Method of Euler
74 Errors
75 An Improvement to Euler
76 Higher Order Methods
77 Systems
Numerical Tables
Answers
Index

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