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

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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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