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
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