2 Introduction student for whom he is writing, and since the assumed background and abilities of this model student are sure to have an important in- fluence on how the book gets written, it is only fair that we give you some idea of our own preconceptions about you. We are assuming that, at a mimimum, the usual reader of this book will have completed the equivalent of two years of undergradu- ate mathematics in a U.S. college or university and, in particular, will have had a solid introduction to linear algebra and to multi-variable (aka “advanced”) calculus. But in all honesty, we have in mind some other hoped-for qualities in our reader, principally that he or she is accustomed to and enjoys seeing mathematics presented conceptu- ally and not as a collection of cookbook methods for solving standard exercises. And finally we hope our readers enjoy working out mathe- matical details on their own. We will give frequent exercises (usually with liberal hints) that ask the student to fill in some details of a proof or derive a corollary. A related question is how we expect this book to be used. We would of course be delighted to hear that it has been adopted as the assigned text for many junior and senior level courses in differential equations (and perhaps not surprisingly we would be happy using it ourselves in teaching such a course). But we realize that the book we have written diverges in many ways from the current “standard model” of an ODE text, so it is our real hope and expectation that many students, particularly those of the sort described above, will find it a challenging but helpful source from which to learn about ODEs, either on their own or as a supplement to a more standard assigned text while taking an ODE course. We should mention here—and explain—a somewhat unusual fea- ture of our exposition. The book consists of two parts that we will refer to as “text” and “appendices”. The text is made up of five chap- ters that together contain about two-thirds of the material, while the appendices consist of ten shorter mini-chapters. Our aim was to make the text relatively easy reading by relegating the more diﬃcult and technical material to the appendices. A reader should be able to get a quick overview of the subject matter of one or more chapters by just reading the text and ignoring the references to material in the

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