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Differential Equations: Theory, Technique, and Practice

George F. Simmons and Steven G. Krantz
McGraw Hill
Publication Date: 
Number of Pages: 
The Walter Rudin Student Series in Advanced Mathematics
[Reviewed by
William J. Satzer
, on

Differential Equations: Theory, Technique and Practice is an introductory text in differential equations appropriate for students who have studied calculus. It is based on George Simmons' classic text Differential Equations with Applications and Historical Notes. The preface says that this revised version brings the older text up to date and adds some more timely material while streamlining the exposition in places and augmenting other parts.

While this is a more than adequate introductory book on differential equations, it is rather a disappointment—especially given its heritage. I say this knowing that the revised text is, in several places, pedagogically superior to the original book. Nonetheless, Simmons' classic text has a kind of charm that is still apparent 34 years after its original publication. Its pervading sense of reverence for scholarship and respect for work of past masters were uncommon when the book first appeared and are now rare indeed. Some of Simmons' touch remains, but it seems woefully diluted.

The first two chapters include introductory material and discuss first order ordinary differential equations and second order linear equations. These chapters have several good applications; they retain some of Simmons' original applications and add an attractive new one on the design of a dialysis machine. Simmons' original book began in a way that probably intimidated many students, with a discussion of Picard's existence theorem early on, an analysis of pendulum motion that included talk of elliptic integrals, and an extended example with the brachistochrone. It is understandable that the authors would want to soften the introductory material.

Chapter 3 brings in linear algebra and puts linear differential equations into that context. This is a welcome addition to Simmons' original approach. What follows is a discussion of Picard's existence and uniqueness theorem that doesn't work very well. The authors state the general theorem—after a very brief introduction about Lipschitz conditions—and then they describe the Picard iteration technique and illustrate it with a couple of examples. A well-chosen picture and a few comments about the heuristics of Picard's method would go a long way toward making the theorem and the method seem more plausible. In my experience, simply stating a theorem without proof or motivation is pretty pointless.

To the authors' credit the text preserves most or all of Simmons' original historical notes. However, the authors have added "math nuggets" (short biographical-historical notes) that are sprinkled throughout the text; I found these too short and often distracting. They would be improved, I think, either by fleshing them out more or by providing references for each one.

Each chapter has a concluding section called "Anatomy of an Application". Some of these—for example, the dialysis machine design mentioned earlier and Sturm-Liouville problems in quantum mechanics—work well. Others are not really applications at all: an introduction to the Fourier transform (with no hint of an application thereof), a perturbation method for linear second-order equations, solutions of systems with matrices and exponentials, the Green's function. It is misleading to call these applications when they are simply other related mathematical topics. Surely there are plenty of real applications of differential equations on which the authors could draw. One of the shortcomings of Simmons' original book was the lack of examples from fields outside physics. There are many elementary examples from chemistry, biology, medicine, engineering and other sciences that would serve wonderfully as applications. It's a shame not to include some of them.

Much of the remainder of the book includes topics that parallel Simmons' earlier work. There are chapters on power series solutions, Laplace transforms, the calculus of variations, and systems of first order equations. Added to these in the current volume are chapters on Fourier series, partial differential equations with boundary value problems, and numerical methods. There are two concluding chapters on nonlinear theory and dynamical systems. These are nicely written and another welcome addition to Simmons' original topics. The chapter on numerical methods is rather weak. Tables of data (not especially illuminating themselves) are provided when good graphs—supplemented by tables—might have worked much better. Although the preface to the book mentions computer algebra systems such as MATLAB and Mathematica, the authors do not actually discuss them in the text. There are exercises that ask for computer algebra work, but these seem to be half-heartedly tacked on and are not well thought out.

There are a number of good exercises throughout the book at varying levels of difficulty ranging from basic drill to more challenging problems and exercises involving a more extended exploration of a topic. The "challenge problems" in particular are well-selected, though they tend toward the theoretical. Given the authors' emphasis on modeling in the preface, I would have hoped for more application-related exercises.

The quality of figures throughout the book is rather uneven and ranges from adequate to poor. For example, the plots of vector fields in the first chapter look amateurish and incomplete. The figures in Chapter 9 are also thin on content, and Figure 9.4 is incorrect as drawn. Many of the chapters would benefit from more figures and figures with higher information content.

By and large this is a competent and usable text for a basic course in differential equations. It could have been a lot more.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.



1 What is a Differential Equation?

1.1 Introductory Remarks

1.2 The Nature of Solutions

1.3 Separable Equations

1.4 First-Order Linear Equations

1.5 Exact Equations

1.6 Orthogonal Trajectories and Families of Curves

1.7 Homogeneous Equations

1.8 Integrating Factors

1.9 Reduction of Order

1.9.1 Dependent Variable Missing
1.9.2 Independent Variable Missing

1.10 The Hanging Chain and Pursuit Curves

1.10.1 The Hanging Chain
1.10.2 Pursuit Curves

1.11 Electrical Circuits

Anatomy of an Application: The Design of a Dialysis Machine

Problems for Review and Discovery

2 Second-Order Linear Equations

2.1 Second-Order Linear Equations with Constant Coefficients

2.2 The Method of Undetermined Coefficients

2.3 The Method of Variation of Parameters

2.4 The Use of a Known Solution to Find Another

2.5 Vibrations and Oscillations

2.5.1 Undamped Simple Harmonic Motion
2.5.2 Damped Vibrations
2.5.3 Forced Vibrations
2.5.4 A Few Remarks About Electricity

2.6 Newton’s Law of Gravitation and Kepler’s Laws

2.6.1 Kepler’s Second Law
2.6.2 Kepler’s First Law
2.6.3 Kepler’s Third Law

2.7 Higher Order Linear Equations, Coupled Harmonic Oscillators

Historical Note: Euler

Anatomy of an Application: Bessel Functions and the Vibrating Membrane

Problems for Review and Discovery

3 Qualitative Properties and Theoretical Aspects

3.1 Review of Linear Algebra

3.1.1 Vector Spaces
3.1.2 The Concept Linear Independence
3.1.3 Bases
3.1.4 Inner Product Spaces
3.1.5 Linear Transformations and Matrices
3.1.6 Eigenvalues and Eigenvectors

3.2 A Bit of Theory

3.3 Picard’s Existence and Uniqueness Theorem

3.3.1 The Form of a Differential Equation
3.3.2 Picard’s Iteration Technique
3.3.3 Some Illustrative Examples
3.3.4 Estimation of the Picard Iterates

3.4 Oscillations and the Sturm Separation Theorem

3.5 The Sturm Comparison Theorem

Anatomy of an Application: The Green’s Function

Problems for Review and Discovery

4 Power Series Solutions and Special Functions

4.1 Introduction and Review of Power Series

4.1.1 Review of Power Series

4.2 Series Solutions of First-Order Differential Equations

4.3 Second-Order Linear Equations: Ordinary Points

4.4 Regular Singular Points

4.5 More on Regular Singular Points

4.6 Gauss’s Hypergeometric Equation

Historical Note: Gauss

Historical Note: Abel

Anatomy of an Application: Steady-State Temperature in a Ball

Problems for Review and Discovery

5 Fourier Series: Basic Concepts

5.1 Fourier Coefficients

5.2 Some Remarks about Convergence

5.3 Even and Odd Functions: Cosine and Sine Series

5.4 Fourier Series on Arbitrary Intervals

5.5 Orthogonal Functions

Historical Note: Riemann

Anatomy of an Application: Introduction to the Fourier Transform

Problems for Review and Discovery

6 Partial Differential Equations and Boundary Value Problems

6.1 Introduction and Historical Remarks

6.2 Eigenvalues, Eigenfunctions, and the Vibrating String

6.2.1 Boundary Value Problems
6.2.2 Derivation of the Wave Equation
6.2.3 Solution of the Wave Equation

6.3 The Heat Equation

6.4 The Dirichlet Problem for a Disc

6.4.1 The Poisson Integral

6.5 Sturm-Liouville Problems

Historical Note: Fourier

Historical Note: Dirichlet

Anatomy of an Application: Some Ideas from Quantum Mechanics

Problems for Review and Discovery

7 Laplace Transforms

7.1 Introduction

7.2 Applications to Differential Equations

7.3 Derivatives and Integrals of Laplace Transforms

7.4 Convolutions

7.3.1 Abel's Mechanical Problem

7.5 The Unit Step and Impulse Functions

Historical Note: Laplace

Anatomy of an Application: Flow Initiated by an Impulsively-Started Flat Plate

Problems for Review and Discovery

8 The Calculus of Variations

8.1 Introductory Remarks

8.2 Euler’s Equation

8.3 Isoperimetric Problems and the Like

8.3.1 Lagrange Multipliers
8.3.2 Integral Side Conditions
8.3.3 Finite Side Conditions

Historical Note: Newton

Anatomy of an Application: Hamilton’s Principle and its Implications

Problems for Review and Discovery

9 Numerical Methods

9.1 Introductory Remarks

9.2 The Method of Euler

9.3 The Error Term

9.4 An Improved Euler Method

9.5 The Runge-Kutta Method

Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations

Problems for Review and Discovery

10 Systems of First-Order Equations

10.1 Introductory Remarks

10.2 Linear Systems

10.3 Homogeneous Linear Systems with Constant Coefficients

10.4 Nonlinear Systems: Volterra’s Predator-Prey Equations

Anatomy of an Application: Solution of Systems with Matrices and Exponentials

Problems for Review and Discovery

11 The Nonlinear Theory

11.1 Some Motivating Examples

11.2 Specializing Down

11.3 Types of Critical Points: Stability

11.4 Critical Points and Stability for Linear Systems

11.5 Stability by Liapunov’s Direct Method

11.6 Simple Critical Points of Nonlinear Systems

11.7 Nonlinear Mechanics: Conservative Systems

11.8 Periodic Solutions: The Poincaré-Bendixson Theorem

Historical Note: Poincaré

Anatomy of an Application: Mechanical Analysis of a Block on a Spring

Problems for Review and Discovery

12 Dynamical Systems

12.1 Flows

12.1.1 Dynamical Systems
12.1.2 Stable and Unstable Fixed Points
12.1.3 Linear Dynamics in the Plane

12.2 Some Ideas from Topology

12.2.1 Open and Closed Sets
12.2.2 The Idea of Connectedness
12.2.3 Closed Curves in the Plane

12.3 Planar Autonomous Systems

12.3.1 Ingredients of the Proof of Poincaré-Bendixson

Anatomy of an Application: Lagrange’s Equations

Problems for Review and Discovery