Differential Equations with Applications and Historical Notes: 3rd Edition (Hardback) book cover

Differential Equations with Applications and Historical Notes

3rd Edition

By George F. Simmons

Chapman and Hall/CRC

764 pages | 102 B/W Illus.

CHOICE 2018 Outstanding Academic Title Award Winner
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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.


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

Table of Contents

The Nature of Differential Equations: Separable Equations


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


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


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


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)


Method of Euler


An Improvement to Euler

Higher-Order Methods


About the Author

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).

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations