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

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http://dx.doi.org/10.1090/stml/051/01