Differential Equations & Linear Algebra

Preface PURPOSE AND PREREQUISITES This book is intended as a textbook for a course in differential equations with linear algebra, to follow the differential and integral calculus. Since the syllabus of such a course is by no means standard, we have included more material than can be covered in a single course--possibly enough material for a two-semester course. This additional material is included to broaden the menu for the instructor and to increase the text's subsequent usefulness as a reference book for the student. Written for engineering, science, and computer science students, the approach is aimed at the applications oriented student but is also intended to be rigorous and to reveal the beauty and elegance of the subject. Why blend linear algebra with the differential equations? Since mid-twentieth century, the traditional course in differential equations has been offered in the first or second semester of the sophomore year and has relied on only a minimum of linear algebra, most notably the use of determinants. More recently, beginning with the advent of digital computers on campuses and in industry around the 1960s, a course or part of a course in linear algebra has become a part of most engineering science curricula. Given the current interest in introducing linear algebra earlier in curricula, the growing importance of systems of differential equations, and the natural use of linear algebra concepts in the study of differential equations, it seems best to move toward an integrated approach. FLEXIBILITY The text is organized so as to be flexible. For instance, it is generally considered desirable to include some nonlinear phase plane analysis in a course on differential equations since the qualitative topological approach complements the traditional analytical approach and also powerfully emphasizes the differences between linear and nonlinear systems. However, that topic usually proves to be a "luxury" to which one can devote one or two classes at best. Thus, we have arranged the phase plane material to allow anywhere from a one-class introduction to a moderately detailed discussion: We introduce the phase plane in only four pages in Section 7.3 in support of our discussion of the harmonic oscillator and we return to it in Chapter 11. There, Section 11.2 affords a more detailed overview of the method and provides another possible stopping point. To assist the instructor in the syllabus design we list some sections and subsections as optional but emphasize that these designations are subjective and intended only as a rule of thumb. (To the student we note that "optional" is not intended to mean unimportant, but only as a guide as to which material can be omitted by virtue of not being a prerequisite for the material that follows.) SPECIFIC PEDAGOGICAL DECISIONS Several pedagogical decisions made in writing this text deserve explanation. Chapter sequence:Some instructors prefer to discuss numerical solution early, even within the study of first-order equations. Placement of the material on numerical solution near the end of this text does not rule out such an approach for one could cover Sections 12.1-12.2 on Eider's method, say, at any point in Chapters 2 or 3. Here, it seemed preferable to group Chapters 11 (on the phase plane) and 12 (on numerical solution) together since they complement the analytical approach, the former being qualitative and the latter being quantitative. As such, these two chapters might well have been made the final chapters, with the Laplace transform chapter moving up to precede or to follow Chapter 8 on series solution. Such movement is possible in a course syllabus since other chapters do not depend on series solution or on the Laplace transform. Also along these lines, it might seem awkward that Chapters 4 and 5 on vectors and matrices are separated from Chapter 9 on the eigenvalue problem.Greenberg, Michael D. is the author of 'Differential Equations & Linear Algebra' with ISBN 9780130111180 and ISBN 013011118X.