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Differential Equations with Applications and Historical Notes

George F. Simmons
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

My first course in differential equations was a failure. It may have been my fault, of course. I did learn how to solve linear differential equations, and I remember the endless proof of existence and uniqueness of solutions, particularly the theorem that explained how the local solutions could be assembled into a solution that was valid in as large a region as possible. But very little stuck.

A few years later I found myself needing to teach the basics of differential equations to a class of engineering students, part of their fourth semester calculus course. It was at that point that I ran into George Simmons’s Differential Equations with Applications and Historical Notes and fell in love with it.

Simmons’s book was very traditional, but was full of great ideas, stories, and illuminating examples. Consider, for example, the first chapter, “The Nature of Differential Equations.” After the usual general remarks, Simmons provides examples: families of curves, growth and decay, chemical reactions, falling bodies, and (amazingly) the brachistochrone. (The third edition adds some material from probability theory.) It’s hard going for a beginner, but it immediately establishes the main point: this stuff plays a central role in the physical sciences. Differential equations are powerful.

Simmons’s “historical notes” often appear in footnotes. Some are fairly tame stuff, summaries of the lives of the mathematicians whose names are attached to various equations, methods, or theorems. But every once in a while it gets more exciting. For example, after saying that a certain “surprising property is really quite natural” (on page 275 of this edition) the footnote says “Those readers who are blessed with indomitable skepticism, and rightly refuse to accept assurances of this kind without personal investigation, are invited to consult…” Fantastic! Similarly, after an account (on p. 172) of why the sum of the reciprocals of the squares is \(\pi^2/6\), there is the footnote: “The world is still waiting — more than 200 years later — for someone to discover the sum of the reciprocals of the cubes.” This is a charming book whose author has a clear personality and doesn’t hide his preferences.

Some of my readers may be wondering what the value of \(\zeta(2)\) is doing in a book on differential equations. That is another characteristic of this book: Simmons ranges far and wide through mathematics, bringing in whatever seems useful and relevant to the matter at hand.

The book is not without its faults, as I discovered when I tried to use it with Colby students (more than twenty years ago!). Simmons is mostly interested in physics and engineering, and both his problems and his examples reflect that fact. My class was full of budding economics students who didn’t find this material all that interesting.

The book is definitely old-school. There is some material on qualitative theory of solutions, but there are whole chapters on power series solutions, special functions, Fourier series, and the Laplace transform. Even the calculus of variations gets a chapter. Existence and uniqueness get their chance in the penultimate chapter, followed by a chapter on numerical methods that seems unchanged since the second edition. This is not a competitor for Blanchard-Devaney-Hall or Noonberg.

Some years ago, an attempt was made to update Simmon’s book. The result was published as Differential Equations: Theory, Technique, and Practice, by Simmons and Steven Krantz. Alas, much of the charm of the original disappeared in the new version. So it is good news that CRC has brought back the original book in a third edition. I compared it to the second edition and decided that the changes are mostly minor additions dealing with topics Simmons enjoys. Most importantly, the author’s unique personality shines through.

I wouldn’t use Simmons as the main textbook for a differential equations course. Given the right student, it might make a great source for an independent study. It is definitely worthwhile to have this classic back.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.

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