Xlllcomments. In this text, we also attempt to provide an even balance between
theory, computer solution, and application.
We recommend that the core of a one quarter or one semester course consist
of material from chapters 1, 2, 4, 5, 7, 8 , and sections 10.1, 10 .2, and 10.3. The
remainder of material covered in the course should come from the applications
and models in chapters 3, 6, 9, and sections 10 .4 through 10.11. The selection
of applications and models to be included in the course will depend on the
time available, on the intent of the course, on the student audience, and, of
course, on the preferences of the instructor. The following is a summary of
the material to be found in each chapter.
In Chapter 1 we present a very brief history of the development of calculus
and differential equations. We introduce essential definitions and terminology.
And we define and discuss initial value problems and boundary value prob-
lems.
In Chapter 2 we discuss in detail the first-order initial value problemy' = J(x, y); y(c) = d. First, we define the direction field for the differential
equation y' = f(x, y), we discuss the significance of the direction field, and we
show how to use a computer program to produce a graph of the direction field.
Next, we state a fundamental existence theorem, a fundamental existence and
uniqueness theorem, and a continuation theorem for the initial value problem.
We show how to apply these theorems to a variety of initial value problems
and we illustrate and emphasize the importance of these theorems. Then we
discuss how to find explicit solutions of simple first-order differential equations
such as separable equations and linear equations. Next, we present simple
appli cations of linear first-order differential equations. Finally, we present
some of the simpler single-step, multistep, and predictor-corrector methods
for computing a numerical approximation to the solution of an initial value
problem. We explain how to use a computer program to generate approximate,
numerical solutions to initial value problems. We illustrate and interpret
the various kinds of results which the computer produces. Furthermore, we
reiterate the importance of performing a thorough mathematical analysis,
which includes applying the fundamental theorems to the problem, prior to
generating a numerical solution.
In Chapter 3 we consider a variety of appli cations of the initial value prob-
lem y' = f(x, y); y(c) = d. First, in the section Calculus Revisited, we show
that the solution to the special initial value problem y' = f(x); y(a) = 0 on the
interval [a, b] is equivalent to the definite integral 1: f(x) dx. Then we show
how to use computer software to calculate an approximation to the definite
integral 1: f(x) dx. This will allow one to numerically solve many problems
from calculus. In the sections Learning Theory Models, Population Models,
Simple Epidemic Models, Falling Bodies, Mixture Problems, Curves of Pur-
suit, and Chemical Reactions we examine physical problems from a number