Differential Equations A Systems Approach

9780130460264

0130460265

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 equation