Differential Equations: An Applied Approach

This foundational introduction to modern dynamical systems covers traditional subjects, as well as modern topics such as fundamentals of dynamical systems theory and bifurcation theory. The volume emphasizes analyzing solutions rather than finding solution formulas, introduces numerical methods early in the book and provides case studies for each subject area. Many applications are accompanied with real data and are drawn from many disciplines including many from biological subjects. The volume addresses first order equations, linear first order equations, nonlinear first order equations, systems and higher order equations, homogeneous linear systems and higher order equations, nonhomogeneous linear systems, approximations and series solutions, nonlinear systems and Laplace transforms. For those needing an introduction to modern dynamical systems.

This book is an outgrowth of lecture notes I wrote for an introductory course on differential equations and modeling that I have taught at the University of Arizona for over twenty years. The book offers a blend of topics traditionally found in a first course on differential equations with a coherent selection of applied and contemporary topics that are of interest to a growing and diversifying audience in science and engineering. These topics are supplemented with a brief introduction to mathematical modeling and many applications and in-depth case studies (often involving real data). 'There is enough material and flexibility in the book that an instructor can design a course with any of several different emphases. For example, by appropriate choices of topics one can devise a reasonably traditional course that focuses on algebraic and calculus methods, solution formula techniques, etc.; or a course that has a dynamical systems emphasis centered on asymptotic dynamics, stability analysis, bifurcation theory, etc.; or a course with a significant component of mathematical modeling, applications, and case studies. In my own teaching I strive to strike a balance among these various themes. To do this is sometimes a difficult task, and the balance at which I arrive usually varies from semester to semester, primarily in response to the backgrounds, interests, and needs of my students. Students in my classes have, over the years, come from virtually every college on our campus: sciences, engineering, agriculture, education, business, and even the fine arts. My classes typically include students majoring in mathematics, mathematics education, engineering, physics, chemistry, and various fields of biology.

It is not intended, of course, that all material in the book be covered in a single course. At some schools, some topics in the book might be covered in prerequisite courses and some topics might be taught in other courses. For example, it is now often the case that slope fields, the Euler numerical algorithm, and the separation of variables method for single first-order equations are taught in a first course on calculus. This is the case at the University of Arizona, and therefore I treat these topics as review material. Laplace transforms are not taught in introductory differential courses at the University of Arizona (where they are taught in mathematical methods courses for scientists and engineers), but I do include Laplace transforms in the text because they are an important topic at many schools. All topics in the text are presented in a self contained way and the book can be successfully used with only a traditional first course in (single variable) calculus as a prerequisite.

At some schools linear algebra is prerequisite for a first course in differential equations (or students have had at least some exposure to matrix algebra), while at other schools this is not the case. At yet other schools first courses in linear algebra and differential equations are co-requisite and might even be taught in the same course. A first course in linear algebra is a prerequisite for my course at the University of Arizona. The book, however, develops all topics first without a linear or matrix algebra prerequisite and include follow-up sections that introduce the use of these subjects at appropriate times. I personally have found that my students who have previously studied linear and matrix algebra benefit nonetheless from an introductory presentation without use of these topics, succeeded by a follow-up that brings to play relevant matrix and linear algebraic topics. For a course without a matrix or linear algebra prerequisite, the follow-up sections can simply be omitted, or be used as a brief introduction of these topics (matrix notation and algebra, eigenvalues and eigenvectors, etc.).

A significant component of the course I teach at the University of Arizona consists of applications and case studies. Again, the issue of balance in topics is crucial. A semester or quarter course can reasonably cover only so much material. One option is to include no applications at all, or limit the course application-type examples whose purpose is to illustrate a mathematical point, but to do so in the context of a problem arising in another scientific discipline. Another option is to spend some time on modeling methodology and selected extended applications and case studies. One approach I have used to facilitate the latter goal utilizes semester projects. I typically have each student study a sequence of case studies over the course of the semester, chosen so as to synchronize with the mathematical techniques and procedures developed in the course. For pedagogical reasons I usually have an individual student focus on applications based on a single theme from the same scientific discipline (e.g., population dynamics in biology, objects in motion in physics, etc.). In this way students experience the full modeling procedure, as models are modified and extended in order to meet the challenges of a different set of circumstances and new questions. Other formats are of course possible. For example, one could devise instead a set of case studies that exposes a student to in-depth applications from several different disciplines in order to attain more interdisciplinary breadth.

Another way in which I utilize case studies is to set aside a small number of lectures throughout the course that are devoted to a detailed look at a few selected applications. However I treat applications in a particular course, I do so with three basic goals in mind. One, of course, is to illustrate the use of the mathematical techniques learned in the course. Another goal is to learn some interesting facts about a scientific topic that one obtains by the use of mathematics. For me, just as important as these two goals is the goal of illustrating the way mathematical modeling is done. For this reason when teaching an application I continually make reference to the "modeling cycle" discussed in the Introduction. Although somewhat simplistic, I have found this way of organizing one's thinking about a mathematical application is a useful compass for students when, in the thicket of the details in an extended application, questions arise such as "how do I start?", "what do I do next?", or "why am I doing this?".

For instructors who wish to incorporate some extended applications into their course, a set of case studies appears at the end of each chapter. I chose these studies to illustrate the kinds of equations, the solution and approximation techniques, and analytic methods covered in the chapter. Model derivations are discussed, assumptions are laid out, interpretations and punch lines are drawn from the analysis, and in some cases the results are compared to real data. Many of the applications are presented in a way that applied mathematicians typically work today: exploratory studies are carried out using a computer, conjectures are formulated, and then an attempt is made to corroborate the conjectures using some kind of mathematical analysis and proof.

I like to begin my course with a discussion of what it means to "solve" a mathematical problem and, in particular, a differential equation. Students usually view this question as the search for a solution formula. While solution formulas can be useful, I point out and stress that methods for calculating solution formulas are available for only specialized types of equations. An alternative to a solution formula is a solution approximation. There are many ways to approximate solutions: geometric approaches that approximate graphs of solutions, numerical methods that estimate solution values, formulas for solution approximations (e.g., Taylor polynomial approximations), and methods that approximate the equation by a simpler equation (e.g., the linearization procedure). All of these approximation procedures appear in the book (and are used in the applications). In an introductory course, it is of course also important to learn about some special types of equations for which methods are available for the calculation of solution formulas. There are several reasons for this. Equations of these specialized types do sometimes arise in applications and their solution formulas can be useful. Some special types of equations (e.g., linear equations) often serve as approximations to more complicated equations; however, an approximating equation is useful only if it is more tractable in some way than the original equation (e.g., a solution formula is available). Also, solvable types of equations serve as "targets" for transformations, that is, a change of variable might transform an equation to a special type for which a solution formula can be found. Finally, special kinds of equations serve useful pedagogical purposes as aids in learning about and understanding differential equations. For this reason the text covers several of the most important types of specialized differential equations and procedures for the calculation of their solution formulas.

Organization

The book begins with the study of a single first-order equations (in Chapters 1, 2, and 3) and then covers (in Chapter 4 through 9) systems of first order equations (which includes higher order equations). The treatment of systems deliberately parallels that of first-order equations. It is useful for the instructor to keep this in mind while covering the material in Chapters 1, 2 and 3 on first order equations and to take advantage of having introduced various concepts, definitions, theorems and methods in the first order case when it comes time to cover systems and higher order equations in later chapters. Examples include the fundamental existence and uniqueness theorem, slope and vector field analysis, numerical approximations methods, the structure of the general solution of linear equations, phase plane analysis, equilibria categorization and stability analysis, bifurcation theory, and series methods of approximation.

Chapter 1 begins with a fundamental existence and uniqueness theorem for initial value p...