Introduction This book is about differential equations—a very big subject! It is so extensive, in fact, that we could not hope to cover it completely even in a book many times this size. So we will have to be selective. In the first place, we will restrict our attention almost entirely to equations of evolution. That is to say, we will be considering quanti- ties q that depend on a “time” variable t, and we will be considering mainly initial value problems. This is the problem of predicting the value of such a quantity q at a time t1 from its value at some (usu- ally earlier) “initial” time t0, assuming that we know the “law of evolution” of q. The latter will always be a “differential equation” that tells us how to compute the rate at which q is changing from a knowledge of its current value. While we will concentrate mainly on the easier case of an ordinary differential equation (ODE), where the quantity q depends only on the time, we will on occasion consider the partial differential equation (PDE) case, where q depends also on other “spatial variables” x as well as the time t and where the partial derivatives of q with respect to these spatial variables can enter into the law determining its rate of change with respect to time. Our principal goal will be to help you develop a good intuition for equations of evolution and how they can be used to model a large variety of time-dependent processes—in particular those that arise in the study of classical mechanics. To this end we will stress various metaphors that we hope will encourage you to get started thinking creatively about differential equations and their solutions. But wait! Just who is this “you” we are addressing? Every text- book author has in mind at least a rough image of some prototypical 1
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