Differential Equations: Applied Approach by J. M. Cushing

• Table of Contents
• Forewords & Introductions

Table of Contents

Forewords & Introductions

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, andvarious 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 problems associated with a single first order equation. This general theorem is followed by graphical and numerical approximation procedures for solutions. These approximation methods appear early in the book because they are general (i.e., are not limited to specialized types of equations) and are therefore available for use throughout the book. Chapters 2 and 3 study solution formula methods, for selected special types of first order equations, and some basic qualitative and approximation methods of analyzing solutions appropriate when formulas are unavailable (or unnecessary). Chapter 2 treats linear equations and the integrating factor method, which results in the fundamentally important Variation of Constants formula in the one dimensional (scalar) case. After!a discussion of some shortcut solution methods (e.g., Undetermined Coefficients), Chapter 2 closes with a brief look at autonomous linear equations. Although rather tAvial from the point of view of solution formulas, these equations introduce (in the simplest setting) some fundamental concepts of dynamical systems (equilibria, stability, phase line portraits, etc.) that anticipate some of the main themes of Chapter 3.
Nonlinear equations are the subject of Chapter 3. The chapter begins (Section 1) with a fairly thorough introduction to the qualitative analysis of phase line portraits for autonomous equations, including an introduction to bifurcation theory. This analysis illustrates how one can obtain great deal of information about solutions of a differential equation without use of solution formulas. Autonomous equations are, however, a special type of separable equation, a category of equations for which we can in principle obtain solution formulas (provided the necessary integrals can be calculated). Separable equations are covered in Section 2. Section 3 deals with a selection of other special types of equations and the methods for finding solution formulas. The main theme of this section is how an appropriate change of variable can transform an equation to a type for which a solution method is known. Section 4 presents some analytic approximation methods for first order equations, including Taylor polynomial and Picard iteration methods. Two forms of Taylor polynomial approximations are given. It is common in an introductory course to cover solution approximation formulas using Taylor polynomials in the independent variable (and the limiting case of power series formulas for solutions). Section 4 also includes a method of approximation based on Taylor polynomials in a parameter that appears in the equation. This is not so commonly done in introductory courses, but I find quite natural and straightforward in my course to introduce and use this example of a "perturbation" method. (Perturbation methods are of great importance in both classical and modern applied mathematics.)
Chapters 4 through 8 cover systems of first order equations (and higher order equations). The development parallels that of single first order equations in Chapters 1 through 3. The fundamental existence and uniqueness theorem for initial values problems, and the graphical and numerical approximations studied in Chapter 1, are extended to systems of equations in Chapter 4. Chapters 5 and 6 cover linear systems. Homogeneous linear systems are studied in Chapter 5, with an emphasis placed on the solution of autonomous systems and their phase plane portraits. Solution procedures for single linear equations studied in Chapter 2 are extended to nonhomogeneous systems of linear equations in Chapter 6 (specifically, the Variation of Constants formula and the method of Undetermined Coefficients). The approximation techniques introduced for single first order equations in Chapter 3 (Taylor polynomial, Picard, and perturbation approximations) are extended to systems in Chapter 7. Higher order equations are given special treatment in Chapters 5, 6 and 7 in their own sections. Nonlinear systems (and higher order equations) are the subject of Chapter 8, which focuses primarily on planar autonomous systems and the problem of constructing phase plane portraits. The basic notions of equilibria, stability, linearization, and bifurcation first introduced in Chapter 3 are extended to systems in Chapter 7 (which also includes an introduction to Poirrcar-Bendixson theory). The focus throughout is on two dimensional systems, although closing sections briefly discuss extensions to higher dimensional systems. The book concludes with an study of Laplace transforms for linear equations (first order, higher order, and systems).
A typical syllabus I use in my course is:
Chapter 1, Chapter 2, Chapter 3.1, 3.4, Chapter 4, Chapter 5.1-5.5, Chapter 6, Chapter 8.