You are here

A Course in Differential Equations with Boundary-Value Problems

Stephen A. Wirkus, Randall J. Swift, and Ryan Szypowski
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
, on

This voluminous introduction to differential equations has material sufficient to support a variety of undergraduate courses from basic to more advanced levels. It appears to have been designed to offer instructors considerable flexibility in selecting topics to meet the goals of a particular course. It also provides options to accommodate academic schedules that range from individual quarters to multiple semesters.

The authors follow a traditional approach to ordinary differential equations with a few wrinkles and a touch of a more modern approach. They begin with first order linear equations and continue to treat linear equations and linear systems completely before introducing nonlinear systems. Numerical methods are introduced early in the book. Parallel computer lab exercises are provided at the end of each chapter in Matlab, Mathematica, and Maple. An appendix provides short tutorials for each of these software packages. The authors also introduce some qualitative analysis early, largely in the form of direction fields.

All the linear algebra that’s needed is presented early in the chapter that introduces systems of differential equations. The authors explore qualitative analysis of systems by introducing the phase plane and illustrating its use with the harmonic oscillator system. They go on to describe the various possible equilibria that one can encounter and expand that to consider nonlinear systems and bifurcations in systems that depend on a parameter. A series of examples with epidemiological and ecological models demonstrate the usefulness of these ideas in applications.

Laplace transforms and power series methods for solving differential equations are treated in mostly standard ways. A chapter on boundary value problems (BVPs) touches on several areas including two-point BVPs, Sturm-Liouville theory, orthogonal functions, and Fourier series. It is essentially a quick survey that gives students a taste of the subject. Likewise, the chapter on partial differential equations is a relatively short introduction to the subject concentrating on the heat, wave and Laplace equations.

Besides the exercises and computer labs that are presented in each chapter, the authors also provide projects that explore some subjects in greater detail or introduce new related material. These include applications (like signal processing, integral equations, and stochastic processes) as well as more advanced mathematical material (e.g., hypergeometric functions, Chebyshev polynomials).

The book’s flexible organization offers instructors a lot of options for courses of various lengths, levels, and choices of computer software. This offers some obvious advantages. But it also makes for a very long book and one that may not be well matched to any particular course. Yet it is thoughtfully assembled, well written and attentive to the needs of students.

This text has much in common with another book with some of the same authors. The first edition of that book was reviewed here. The current book, itself a second edition, shares eight identical chapters and two appendices with the second edition of the prior book. The book reviewed here adds the two chapters on boundary value problems and partial differential equations.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Traditional First-Order Differential Equations

Introduction to First-Order Equations

Separable Differential Equations

Linear Equations

Some Physical Models Arising as Separable Equations

Exact Equations

Special Integrating Factors and Substitution Methods

Bernoulli Equation

Homogeneous Equations of the Form g(y=x)

Geometrical and Numerical Methods for First-Order Equations

Direction Fields|the Geometry of Differential Equations

Existence and Uniqueness for First-Order Equations

First-Order Autonomous Equations|Geometrical Insight

Graphing Factored Polynomials

Bifurcations of Equilibria

Modeling in Population Biology


Numerical Approximation: Euler and Runge-Kutta Methods

An Introduction to Autonomous Second-Order Equations

Elements of Higher-Order Linear Equations

Introduction to Higher-Order Equations

Operator Notation

Linear Independence and the Wronskian

Reduction of Order|the Case n = 2

Numerical Considerations for nth-Order Equations

Essential Topics from Complex Variables

Homogeneous Equations with Constant Coe□cients

Mechanical and Electrical Vibrations

Techniques of Nonhomogeneous Higher-Order Linear Equations

Nonhomogeneous Equations

Method of Undetermined Coe□cients via Superposition

Method of Undetermined Coe□cients via Annihilation

Exponential Response and Complex Replacement

Variation of Parameters

Cauchy-Euler (Equidimensional) Equation

Forced Vibrations

Fundamentals of Systems of Differential Equations

Useful Terminology

Gaussian Elimination

Vector Spaces and Subspaces

The Nullspace and Column Space

Eigenvalues and Eigenvectors

A General Method, Part I: Solving Systems with Real and Distinct or Complex


A General Method, Part II: Solving Systems with Repeated Real Eigenvalues

Matrix Exponentials

Solving Linear Nonhomogeneous Systems of Equations

Geometric Approaches and Applications of Systems of Differential Equations

An Introduction to the Phase Plane

Nonlinear Equations and Phase Plane Analysis

Systems of More Than Two Equations


Epidemiological Models

Models in Ecology

Laplace Transforms


Fundamentals of the Laplace Transform

The Inverse Laplace Transform

Laplace Transform Solution of Linear Differential Equations

Translated Functions, Delta Function, and Periodic Functions

The s-Domain and Poles

Solving Linear Systems Using Laplace Transforms

The Convolution

Series Methods

Power Series Representations of Functions

The Power Series Method

Ordinary and Singular Points

The Method of Frobenius

Bessel Functions

Boundary-Value Problems and Fourier Series

Two-Point Boundary-Value Problems

Orthogonal Functions and Fourier Series

Even, Odd, and Discontinuous Functions

Simple Eigenvalue-Eigenfunction Problems

Sturm-Liouville Theory

Generalized Fourier Series

Partial Differential Equations

Separable Linear Partial Differential Equations

Heat Equation

Wave Equation

Laplace Equation

Non-Homogeneous Boundary Conditions

Non-Cartesian Coordinate Systems

A An Introduction to MATLAB, Maple, and Mathematica


Some Helpful MATLAB Commands

Programming with a script and a function in MATLAB


Some Helpful Maple Commands

Programming in Maple


Some Helpful Mathematica Commands

Programming in Mathematica

B Selected Topics from Linear Algebra

A Primer on Matrix Algebra

Matrix Inverses, Cramer's Rule

Calculating the Inverse of a Matrix

Cramer's Rule

Linear Transformations

Coordinates and Change of Basis

Similarity Transformations

Computer Labs: MATLAB, Maple, Mathematica