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About the book From the preface of: Differential Equations, Fourth 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. At Boston University, we decided to revise our course, and we wrote this book to support our efforts. We now approach our 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 (laptops, ipads, smart phones.) that we carry around with us everywhere. 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 we can give our students a glimpse of the nature of contemporary mathematical research. The Qualitative, Numeric, and Analytic Approaches Accordingly, this book is a very different from the typical 'cookbook' differential equations text. We have eliminated many of the 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 easily studied using a computer, we also emphasize numerical techniques. Building The Dream Gwendolyn Wright Pdf Writer more.
DETools, the software that accompanies this book, provides students with ample computational tools to investigate the behavior of solutions of differential equations both numerically and graphically. 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, we do the calculations. But we always examine the resulting formulas using qualitative and numerical points of view as well. How This Book is Different 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 advocate a dynamical systems point of view. That is, 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. 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 vary. It is our philosophy that using a computer is as natural and necessary to the study of differential equations as is the use of paper and pencil. DETools should make the inclusion of technology in the course as easy as possible. This suite of computer programs illustrates the basic concepts of differential equations. Three of these programs are solvers which allow the student to compute and graph numerical solutions of both first-order equations and systems of differential equations.