From the preface of:

Differential Equations, First Edition, by Blanchard, Devaney, Hall.

The study of differential equations is a beautiful application of the ideas and techniques of calculus to our everyday lives. Indeed, it could be said that calculus was developed mainly so that the fundamental principles that govern many phenomena could be expressed in the language of differential equations. Unfortunately, it was difficult to convey the beauty of the subject in the traditional first course on differential equations because the number of equations that can be treated by analytic techniques is very limited. Consequently, the course tended to focus on technique rather than on concept.

This book is an outgrowth of our opinion that we are now able to effect a radical revision, and we approach our updated course with several goals in mind. First, the traditional emphasis on specialized tricks and techniques for solving differential equations is no longer appropriate given the technology that is readily available. Second, many of the most important differential equations are nonlinear, and numerical and qualitative techniques are more effective than analytic techniques in this setting. Finally, the differential equations course is one of the few undergraduate courses where it is possible to give students a glimpse of the nature of contemporary mathematical research.

The Qualitative, Numeric, and Analytic Approaches

Accordingly, this book is a radical departure from the typical "cookbook" differential equations text. We have eliminated most specialized techniques for deriving formulas for solutions, and we have replaced them with topics that focus on the formulation of differential equations and the interpretation of their solutions. To obtain an understanding of the solutions, we generally attack a given equation from three different points of view.

One major approach we adopt is qualitative. We expect students to be able to visualize differential equations and their solutions in many geometric ways. For example, we readily use slope fields, graphs of solutions, vector fields, and solution curves in the phase plane as tools to gain a better understanding of solutions. We also ask students to become adept at moving among these geometric representations and more traditional analytic representations.

Since differential equations are readily studied using the computer, we also emphasize numerical techniques. We assume that students have access to some sort of technology that approximates solutions and graphs these solutions easily. Even if we can find an explicit formula for a solution, we often work with the equation both numerically and qualitatively to understand the geometry and the long-term behavior of solutions. When we can find explicit solutions easily (such as in the case of separable first-order equations or constant-coefficient, linear systems), we do the calculations. But we never fail to examine the resulting formulas we obtain using qualitative and numerical points of view as well.

Specific Changes

There are several specific ways in which this book differs from other books at this level. First, we incorporate modeling throughout. We expect students to understand the meaning of the variables and parameters in a differential equation and to be able to interpret this meaning in terms of a particular model. Certain models reappear often as running themes, and they are drawn from a variety of disciplines so that students with various backgrounds will find something familiar.

We also adopt a dynamical systems point of view. Thus, we are always concerned with the long-term behavior of solutions of an equation, and using all of the appropriate approaches outlined above, we ask students to predict this long-term behavior of solutions. In addition, we emphasize the role of parameters in many of our examples, and we specifically address the manner in which the behavior of solutions changes as these parameters are varied.

Like other texts, we begin with first-order equations, but the only analytic technique we use to find closed-form solutions is separation of variables (and, at the end of the chapter, an integrating factor or two to handle certain linear equations). Instead, we emphasize the meaning of a differential equation and its solutions in terms of its slope field and the graphs of its solutions. If the differential equation is autonomous, we also discuss its phase line. This discussion of the phase line serves as an elementary introduction to the idea of a phase plane, which plays a fundamental role in subsequent chapters.

We then move directly from first-order equations to systems of first-order differential equations. Rather than consider second-order equations separately, we convert these equations to first-order systems. When these equations are viewed as systems, we are able to use qualitative and numerical techniques more readily. Of course, we then use the information about these systems gleaned from these techniques to recover information about the solutions of the original equation.

We also begin the treatment of systems with a general approach. We do not immediately restrict our attention to linear systems. Qualitative and numerical techniques work just as easily when a system is nonlinear, and one can proceed a long way toward understanding systems without resorting to algebraic techniques. However, qualitative ideas do not tell the whole story, and we are led naturally to the idea of linearization. With this background in the fundamental geometric and qualitative concepts, we then discuss linear systems in detail. As always, we not only emphasize the formula for the general solution of a linear system but also the geometry of its solution curves and of the related eigenvectors and eigenvalues.

While our study of systems requires the minimal use of some linear algebra, it is definitely not a prerequisite. As we deal primarily with two-dimensional systems, we easily develop all of the necessary algebraic techniques as we proceed. In the process, we give considerable insight into the geometry of such topics as eigenvectors and eigenvalues.

These topics form the core of our approach. However, there are many additional topics that one would like to cover in the course. Consequently, we have included discussions of forced second-order equations, nonlinear systems, Laplace transforms, numerical methods, and discrete dynamical systems. Although some of these topics are quite traditional, we always present them in a manner that is consistent with the philosophy developed in the first half of the text.

At the end of each chapter, we have included several "labs." Doing detailed numerical experimentation and writing reports has been our most successful modification of the traditional course at Boston University. Good labs are tough to write and to grade, but we feel that the benefit to students is extraordinary.

Pathways Through This Book

There are a number of possible tracks that instructors can follow in using this book. We feel that Chapters 1-3 form the core (with the possible exception of Sections 2.5 and 3.8, which cover systems in three dimensions). Most of the later chapters assume familiarity with this material. Certain sections such as Section 1.7 (bifurcations) and Section 1.9 (changing variables) may be skipped if some care is taken in choosing material from subsequent sections. However, the material on phase lines and phase planes, qualitative analysis, and solutions of linear systems is central.

A typical track for an engineering-oriented course would follow Chapters 1-3 (perhaps skipping Sections 1.9, 2.5, and 3.8). These chapters will take roughly two-thirds of a semester. The final third of the course might cover Sections 4.1-4.3 (forced, second-order linear equations and resonance), Section 5.1 (linearization of nonlinear systems), and Chapter 6 (Laplace transforms). Chapters 4 and 5 are independent of each other can be covered in either order. In particular, Section 5.1 on linearization of nonlinear systems near equilibrium points forms an excellent capstone for the material on linear systems in Chapter 3.

Incidentally, it is possible to cover Sections 6.1 and 6.2 (Laplace transforms for first-order equations) immediately after Chapter 1. As we have learned from our colleagues in the College of Engineering at Boston University, some engineering programs teach a circuit theory course that uses the Laplace transform at an earlier point than is typically the case. Consequently, Sections 6.1 and 6.2 are written so that the differential equations course and such a circuits course could proceed in parallel. However, if possible, we recommend waiting to cover Chapter 6 entirely until after the material in Sections 4.1-4.3 has been discussed.

Instructors may wish to substitute material on discrete dynamics (Chapter 8) for Laplace transforms. A course for students with a strong background in physics might involve more of Chapter 5, including a treatment of Hamiltonian (Section 5.3) and gradient systems (Section 5.4). A course geared toward applied mathematics might include a more detailed discussion of numerical methods (Chapter 7).

Changes in the First Edition

We have been quite pleased with the reception that the preliminary edition of this book has enjoyed since its publication in 1995. We are especially indebted to the large number of readers and instructors who made comments about various points in the earlier edition. Accordingly, we have made some changes in this edition. The most significant changes include more thorough treatments of forcing and resonance for second-order equations and a revised treatment of Laplace transforms. The material in Chapter 2 has been extensively rewritten to follow more closely our intent to introduce analytic, qualitative, and numerical methods for systems at an early stage. Two appendices have been added. The first is an alternate treatment of first-order linear equations and can be used in place Section 1.8. The second appendix is a review of complex numbers and Euler's formula.

Most of the other changes involve only minor rearrangements of topics so that most instructors can avoid skipping sections within a chapter. As with any significant revision of an existing course, we anticipate that this book will continue to evolve in future editions. We encourage comments, suggestions, and criticism. The best way to comment is to send email to odes@math.bu.edu. We'll do our best to acknowledge the email, but we will definitely read and consider every comment.

Our Website and Ancillaries

Readers and instructors are invited to make extensive use of our web site

Our publisher, Brooks/Cole, also maintains the DiffEQ Resource Center at

The Boston University Differential Equations Project

This book is a product of the now complete National Science Foundation Boston University Differential Equations Project sponsored by the National Science Foundation (NSF Grant DUE-9352833) and Boston University. The goal of that project was to rethink the traditional, sophomore-level differential equations course. We are especially thankful for that support.

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