Differential Equations With Boundary Value Problems

The introductory differential equations course plays an interesting role in the undergraduate mathematics curriculum. It is a required course for most science and engineering students, many of whom will take major courses that require certain knowledge of and skills related to differential equations. The course also plays the role of an introductory "applied course" for mathematics students, where modeling is often a primary focus. Our point of view, which in no way diminishes the utility of the course's content, is that the differential equations course has two overarching functions: (1) It is where students are introduced to the central subject in all of applied mathematics, a subject whose questions have spawned most of the classical theory of analysis since the time of Newton and continue to create a rich field of mathematical activity today. (2) It is a place to reinforce and extend the student's (perhaps tenuous) understanding of calculus. It is the next step after calculus on the path that leads to real analysis. Whether the student will follow that path any farther makes little difference. Throughout this book, we attempt to emphasize that the primary goal when studying a differential equation is always to understand the behavior of its solutions. Writing down formulas for solutions is just one means to that end. Qualitative and numerical methods are equally, if not more, important. Moreover, all of these "tools" are manifestations of theory, and we endeavor to emphasize that fact at a level which is appropriate for the course. Times have changed. Amazingly fast computers are commonplace, and powerful software systems such as Mathematica, Maple, and MATLAB make it possible to solve highly complex problems and create wonderfully illuminating graphics. This computing power has enormous potential for enhancing the differential equations course-perhaps more so than in any other mathematics course. Indeed, the computer can facilitate the analysis of solutions algebraically, numerically, and graphically. Yet the computer cannot teach the conceptual and theoretical foundations that this book strives to convey-but neither can tedious algebraic manipulations done with paper and pencil. This book does not emphasize technology, and only occasionally do we mention it directly. However, we recommend the use of a computer to facilitate those computations that require more than a few routine algebraic steps. We are platform-neutral with respect to technology, and our belief is that the best way to address technology specifically is with separate companion manuals. Thus we have created A Mathematica Companion for Differential Equations and A Maple Companion for Differential Equations for those who use Mathematica or Maple in their courses, and we highly recommend their use. Organization and Content For the most part, this book covers the traditional topics in the introductory, differential equations course, while enhancing the usual algebraic approach with more geometric ideas and interpretations. Naturally reflecting the biases of the author (an inkling of which should be provided by the preceding commentary), certain standard topics are omitted or given less emphasis, while others are given increased emphasis. A few of the topics covered are rather novel for an introductory text. The following are some of our more significant deviations from the "standard course": Systems and numerical methods are introduced early. Series methods are deemphasized, particularly the method of Frobenius. The method of undetermined coefficients is banished to near oblivion by the use of "exponential shift" and complex solutions. Linear equations of order three or higher are relegated to an appendix. An entire chapter is devoted to nonlinear systems in applications. We personally believe that this is where the subject of differential equations really becomes exciting and that sHollis, Selwyn L. is the author of 'Differential Equations With Boundary Value Problems' with ISBN 9780130159274 and ISBN 0130159271.