Differential Equations: A Systems Approach

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9780130460264: Differential Equations: A Systems Approach

Written for beginners, this well organized introduction promotes a solid understanding of differential equations that is flexible enough to meet the needs of many different disciplines. With less emphasis on formal calculation than found in other books all the basic methods are covered—first order equations, separation, exact form, and linear equations—as well as higher order cases, linear equation with constant and variable coefficients, Laplace transform methods, and boundary value problems. The book's systems focus induces an intuitive understanding of the concept of a solution of an initial value problem in order to resolve potential confusion about what is being approximated when a numerical method is used. The author outlines first order equations including linear and nonlinear equations and systems of differential equations, as well as linear differential equations including the Laplace transform, and variable coefficients, nonlinear differential equations, and boundary problems and PDEs. For those looking for a solid introduction to differential equations.

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If an application of mathematics has a component that varies continuously as a function of time, then it probably involves a differential equation. For this reason, ordinary differential equations are of great importance in engineering, applied mathematics and the sciences. This has been recognized since the founders of calculus, Newton and Leibniz, made their contributions to the subject in the late seventeenth century.

A differential equations text has to address an audience with diverse interests. Science and engineering majors are required to take a differential equations course because it provides them with valuable mathematical tools. Mathematics majors may take courses in differential equations because the subject is interesting; because it is an essential component of applied mathematics; or because it is prerequisite for the study of differential geometry, dynamical systems, and mathematical modeling. All students who take a differential equations course will gain a deeper understanding of the concepts and applications of calculus.

There are numerous differential equations texts on the market, and it is reasonable to ask why I chose to write another one. I had taught many differential equations courses before I decided to write this text, and what tipped the scale was a conversation overheard in the mathematics library between two students. The gist of the conversation was that the differential equations course was trivial, which may be true if the course focuses only on the formalities of solving equations for which algorithms are available. If viewed as an application of the quadratic formula, solving second-order linear equations with constant coefficients, for example, is easy. A differential equations course, like any mathematics course, needs to offer more intellectual challenge than that. I wanted to write a text that will enable students to visualize a differential equation as a direction or vector field, and to use the standard formal solution procedures with a full understanding of their limitations.

This text maintains a moderate level of rigor. Proofs are included if they are accessible and have the potential to enhance the reader's understanding of the subject. For example, the existence theorem for solutions of initial value problems is not actually proved—its prerequisite, Ascoli's theorem, would not ring a bell for most readers—but I allude to Peano's proof, which uses Eider's method to show that a solution exists. On the other hand, the proof of the uniqueness theorem is included, as a special case of Proposition 2.4.3, which specifies an upper bound for the rate at which solutions of a differential equation can diverge from one another.

Although applications usually involve systems of differential equations, the emphasis in most differential equations texts is second-order equations. When faced with a system, there is a rather complicated algorithm that finds an equivalent higher-order equation. This approach doesn't carry one very far, and it stems from a desire to avoid linear algebra in general and characteristic roots (eigenvalues) in particular. In fact, there is no result presented in the introductory linear algebra course that is not useful in differential equations, and linear algebra ought to be a prerequisite for the differential equations course. In spite of this, many universities, including the one that employs me, require only two semesters of calculus as a prerequisite for their differential equations courses. There are texts that present differential equations and linear algebra as a combined course. This is an acceptable approach, to which this text is an alternative. I have attempted to accommodate the needs of readers who have not had a linear algebra course, without wasting the time of those who have had the course. Thus, Chapter 4 presents linear systems of differential equations in matrix form but is limited to systems of two equations—and thus involves only 2 x 2 matrices. The time spent making this accessible to those who have not had a linear algebra course is not excessive. Linear operators are defined and discussed in Chapter 5, along with further concepts from linear algebra: linear combinations and linear independence.

We should tell students about IVP (initial value problem) solvers. These are computer programs that can calculate and plot an approximate solution of an initial value problem. People who work with differential equations find them indispensable; yet many students complete a differential equations course without ever using one. It is common to find the IVP solver algorithms in differential equations texts. The Runge-Kutta algorithm is often found, for example. These algorithms should be studied in numerical analysis courses, but our job is to get the students to use them. A solution of a differential equation that was not obtained by symbol manipulation is for many people a first encounter with a function that is not presented as a formula. While no student should pass a differential equations course without learning how to solve certain differential equations by analytic means, students must be trained to use numerical methods as well. My goal has been to induce an intuitive understanding of the concept of a solution of an initial value problem in order to resolve potential confusion about what we are approximating when we call a numerical method. This is not a numerical analysis text, and the discussion of numerics is confined to Euler's method—which advances the understanding of what a differential equation is—and a brief user's guide to more advanced methods. I do not hesitate to call upon effective numerical methods when symbolic methods can't be used.

Technology and Supplements

It is widely believed that computer algebra software (CAS) can make short work of the routine calculations that have bedeviled generations of students in introductory differential equations courses. I have found that this software is sometimes beneficial as a laborsaving device, and that it is definitely useful for producing illustrations. There are three admirable CAS programs available: Maple, Mathematica, and MATLAB. Most of the illustrations in this text were produced with Mathematica. Examples using CAS programs can be found on the text Web site, http://www.prenhall.com/conrad. This text can be used effectively without the benefit of CAS, but an IVP solver that will display graphs of solutions of differential equations on a computer or calculator screen is required. Every CAS can function as an IVP solver, and special-purpose IVP solvers may be downloaded by following links on the text Web site. The Web site also has a list of currently available hand-held graphing calculators that include IVP solvers. There is a general discussion of NP solvers in Section 2.3.

For those who wish to use Maple with this text, there is a new manual, Maple Projects for Differential Equations (013-047974-8), by Gilbert and Hsiao, that can be shrink wrapped with the text for one half the manual's normal price.

The same is also true for Polking and Arnold's Ordinary Differential Equations Using Mf1TLAB (013-011381-6). This text also is accompanied by a Student Solutions Manual and an Instructor Solutions Manual.

Organization

To narrow the field when selecting a differential equations text, an instructor may ask if the applications are realistic. This text does not favor applications involving "real data." While there is much to be said for studying realistic applications, the many complications may obscure the differential equation that should be the center of attention. It is preferable to consider a simple problem that exemplifies the differential equations component of an actual application.

The existence and uniqueness theorems for initial value problems have implications about the structure of the set of solutions of a differential equation or system. This text explores that structure, as well as the additional structural properties of special kinds of equations, such as linear equations or autonomous systems have. The properties of linearity and of being autonomous are important for many applications, and by associating these properties with particular applications, we can bring our physical experience to the task of learning about differential equations.

This text is presented in three parts: The first introduces both linear differential equations and nonlinear systems and provides a foundation. The second part is devoted to linear differential equations, including systems of first-order equations, the single second-order equation, Laplace transform methods, and equations with variable coefficients. The third part focuses on nonlinear differential equations and dynamical systems. The dependency graph that follows the table of contents will be of assistance in navigating the text and in planning class syllabi. In this graph, the Sections marked with asterisks in the table of contents are treated as separate nodes. They can be skipped without interruption of the sequence within the Chapters where they reside. In a recent course at Temple University, given over a semester of fourteen weeks (not including the final examination), I covered Chapters 1-6, and Section 8.1, skipping Sections 1.5, 2.5, 3.4, 3.5, 5.9, and 6.8. The class met three hours per week, with an additional weekly Maple-based computer laboratory.

I have used an approach to definitions that is more common in lower-level mathematics texts. Instead of placing formal, numbered statements of definitions in the text, I have provided each Chapter with a glossary that contains all of the definitions, in alphabetical order. This allows for a more informal discussion of a term when it is introduced. The first use of a term that is defined in the glossary is in boldface. I have reserved the bold typeface for that purpose—except when it is used in Section headings and mathematical expressions. My hope is th...

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