You are here

Differential Equations and Linear Algebra: Computing and Modeling

C. Henry Edwards and David E. Penney
Prentice Hall
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

This review considers two textbooks on differential equations that are designed for a course following calculus: Differential Equations and Linear Algebra and Differential Equations: Computing and Modeling, both by Edwards and Penney. The two books are similar in many ways, and some chapters are essentially identical. The differences are due to a divergence in aims for the two books that is largely reflected in their titles. One of the books incorporates an introduction to linear algebra and integrates linear algebra with its treatment of differential equations. The other book emphasizes computation and modeling more extensively and introduces a limited amount of linear algebra, largely in the context of systems of linear differential equations. There are also versions of this book that include boundary value problems.

Both of these books have been extensively classroom-tested and modified in succeeding editions based on classroom experience. The authors’ general goals in both books are the same: to concentrate on conceptual development and to use applications to provoke and maintain student engagement. An application manual, available with both textbooks, provides descriptions of how Maple, Mathematica, and MATLAB can support numerical and symbolic investigations of differential equations.

As previously noted, the two books have a lot of material in common. Chapters in common include: first order equations, mathematical models and numerical methods, higher order equations, linear systems of differential equations, nonlinear systems and Laplace transform methods. Distributed throughout are a number of computing projects called “application modules”, and many of these are the same in both books.

Most of the differences between the books consist of additional material in the text with linear algebra. These include the expected chapters on basic linear algebra, as well as chapters on the matrix exponential and on power series solutions.

The books have excellent collections of exercises, and many of these are tied to the application modules. It is especially nice to see so many fresh exercises and so few of the well-worn problems common to many other similar texts. One striking example is a set of exercises dealing with earthquake-induced vibrations of multi-story buildings that is a terrific application of second order linear systems.

Another strength of the books is that the authors, having chosen to treat numerical methods, do so with care and in detail. Of course, not all introductory differential equations texts need to include numerical methods, but when they do it should be more than perfunctory. So, for example, these books introduce Euler’s method as an algorithm for numerical approximations of solutions of differential equations, explain why numerical methods are sometimes necessary, and then carefully follow up with several examples and then a refined version of the method. After that the authors follow up with an extended discussion of the Runge-Kutta method and explain why it is more accurate than Euler’s method.

Either of these texts would be a worthy choice for an introductory differential equations course. They’d also be a great source of additional exercises and examples for an instructor using a different textbook.

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. First Order Differential Equations.

Differential Equations and Mathematical Models. Integrals as General and Particular Solutions. Slope Fields and Solution Curves. Separable Equations and Applications. Linear First Order Equations. Substitution Methods and Exact Equations.

2. Mathematical Models and Numerical Methods.

Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method, and Improvements. The Runge-Kutta Method.

3. Linear Equations of Higher Order.

Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Nonhomogeneous Equations and Undetermined Coefficients. Forced Oscillations and Resonance. Electrical Circuits. Endpoint Problems and Eigenvalues.

4. Introduction to Systems of Differential Equations.

First-Order Systems and Applications. The Method of Elimination. Numerical Methods for Systems.

5. Linear Systems of Differential Equations.

Linear Systems and Matrices. The Eigenvalue Method for Homogeneous Systems. Second Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems.

6. Nonlinear Systems and Phenomena.

Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. Chaos in Dynamical Systems.

7. Laplace Transform Methods.

Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. Impulses and Delta Functions.