# Differential Equations with Applications and Historical Notes

## Preview

## Book Description

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
- Includes a wealth of unique examples and exercises, as well as the author’s distinctive historical notes, throughout
- Uses explicit explanation to ensure students fully comprehend the subject matter

* *Outstanding Academic Title of the Year, *Choice* magazine, American Library Association.

## Table of Contents

**The Nature of Differential Equations: Separable Equations**Introduction

**General Remarks on Solutions**

**Families of Curves: Orthogonal Trajectories**

**Growth, Decay, Chemical Reactions, and Mixing**

**Falling Bodies and Other Motion Problems**

**Brachistochrone: Fermat and the Bernoullis**

**Miscellaneous Problems for Chapter 1**

**Appendix: Some Ideas from the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation**

**First-Order Equations**Homogeneous Equations

**Exact Equations**

**Integrating Factors**

**Linear Equations**

**Reduction of Order**

**Hanging Chain: Pursuit Curves**

**Simple Electric Circuits**

**Miscellaneous Problems for Chapter 2**

**Second-Order Linear Equations**Introduction

**General Solution of the Homogeneous Equation**

**Use of a Known Solution to Find Another**

**Homogeneous Equation with Constant Coefficients**

**Method of Undetermined Coefficients**

**Method of Variation of Parameters**

**Vibrations in Mechanical and Electrical Systems**

**Newton’s Law of Gravitation and the Motion of the Planets**

**Higher-Order Linear Equations: Coupled Harmonic Oscillators**

**Operator Methods for Finding Particular Solutions**

**Appendix: Euler**

**Appendix: Newton**

**Qualitative Properties of Solutions**Oscillations and the Sturm Separation Theorem

**Sturm Comparison Theorem**

**Power Series Solutions and Special Functions**Introduction: A Review of Power Series

**Series Solutions of First-Order Equations**

**Second-Order Linear Equations: Ordinary Points**

**Regular Singular Points**

**Regular Singular Points (Continued)**

**Gauss’s Hypergeometric Equation**

**Point at Infinity**

**Appendix: Two Convergence Proofs**

**Appendix: Hermite Polynomials and Quantum Mechanics**

**Appendix: Gauss**

**Appendix: Chebyshev Polynomials and the Minimax Property**

**Appendix: Riemann’s Equation**

**Fourier Series and Orthogonal Functions**Fourier Coefficients

**Problem of Convergence**

**Even and Odd Functions: Cosine and Sine Series**

**Extension to Arbitrary Intervals**

**Orthogonal Functions**

**Mean Convergence of Fourier Series**

**Appendix: A Pointwise Convergence Theorem**

**Partial Differential Equations and Boundary Value Problems**Introduction: Historical Remarks

**Eigenvalues, Eigenfunctions, and the Vibrating String**

**Heat Equation**

**Dirichlet Problem for a Circle: Poisson’s Integral**

**Sturm–Liouville Problems**

**Appendix: Existence of Eigenvalues and Eigenfunctions**

**Some Special Functions of Mathematical Physics**Legendre Polynomials

**Properties of Legendre Polynomials**

**Bessel Functions: The Gamma Function**

**Properties of Bessel Functions**

**Appendix: Legendre Polynomials and Potential Theory**

**Appendix: Bessel Functions and the Vibrating Membrane**

**Appendix: Additional Properties of Bessel Functions**

**Laplace Transforms**Introduction

**Few Remarks on the Theory**

**Applications to Differential Equations**

**Derivatives and Integrals of Laplace Transforms**

**Convolutions and Abel’s Mechanical Problem**

**More about Convolutions: The Unit Step and Impulse**

**Functions**

**Appendix: Laplace**

**Appendix: Abel**

**Systems of First-Order Equations**General Remarks on Systems

**Linear Systems**

**Homogeneous Linear Systems with Constant Coefficients**

**Nonlinear Systems: Volterra’s Prey–Predator Equations**

**Nonlinear Equations**Autonomous Systems: The Phase Plane and Its Phenomena

**Types of Critical Points: Stability**

**Critical Points and Stability for Linear Systems**

**Stability by Liapunov’s Direct Method**

**Simple Critical Points of Nonlinear Systems**

**Nonlinear Mechanics: Conservative Systems**

**Periodic Solutions: The Poincaré–Bendixson Theorem**

**More about the Van Der Pol Equation**

**Appendix: Poincaré**

**Appendix: Proof of Liénard’s Theorem**

**Calculus of Variations**Introduction: Some Typical Problems of the Subject

**Euler’s Differential Equation for an Extremal**

**Isoperimetric Problems**

**Appendix: Lagrange**

**Appendix: Hamilton’s Principle and Its Implications**

**The Existence and Uniqueness of Solutions**Method of Successive Approximations

Picard’s Theorem

Systems: Second-Order Linear Equation

**Numerical Methods**

*(by John S. Robertson)*

**Introduction**

**Method of Euler**

**Errors**

**An Improvement to Euler**

**Higher-Order Methods**

**Systems**

## Author(s)

### Biography

**George F. Simmons** has academic degrees from the California Institute of Technology, Pasadena, California; the University of Chicago, Chicago, Illinois; and Yale University, New Haven, Connecticut. He taught at several colleges and universities before joining the faculty of Colorado College, Colorado Springs, Colorado, in 1962, where he is currently a professor of mathematics. In addition to *Differential Equations with Applications and Historical Notes, Third Edition* (CRC Press, 2016), Professor Simmons is the author of *Introduction to Topology and Modern Analysis *(McGraw-Hill, 1963), *Precalculus Mathematics in a Nutshell* (Janson Publications, 1981), and *Calculus with Analytic Geometry* (McGraw-Hill, 1985).

## Reviews

This is an attractive introductory work on differential equations, with extensive information in addition to what can be covered in a two-semester course. The order of the topics examined is slightly unusual in that Laplacians are covered after Fourier transforms and power series. The chapter on power series contains a section on hypergeometric equations, which could well be the first time that an introductory book on the subject goes that far. The book has plenty of exercises at the end of each section, and also at the end of each chapter. The solutions to some of these are included at the end of the book. Most chapters contain a few appendixes that are several pages long. Their subject is either related to the life and work of an exceptional mathematician (such as Newton, Euler, or Gauss) or pertains to an area of mathematics in which the theory of differential equations can be applied. The historical appendixes put the material in context, and explain which parts of the material were the most difficult to discover. The writing is pleasant and reader-friendly throughout. This work is an essential acquisition for all math libraries; no competing works have put the material in such a deep historical context.

--M. Bona, University of Florida