xii- Solve on an interval [a, b] a vector initial value problem consisting of
a system of n first-order differential equations and n initial conditions
where 2 :::; n :::; 6.
Many computer software packages are readily available which include these
features and usually many additional features. Three of the best known and
most widely used packages are MAPLE, Mathematica, and MATLAB®. In
general, each instructor already has his or her own favorite differential equa-
tions software package or combination of packages. For this reason, the text
was written to be independent of any particular software package. The soft-
ware we used to generate solutions and many of the graphs for the examples
as well as the answers to the selected exercises which appear at the end of this
text is contained on the computer disc which accompanies the text. Complete
instructions for running this software are contained in Appendices A and B.It is assumed the reader has completed calculus at least up to and including
the concept of partial derivatives and knows how to add, subtract, multiply,
and divide complex numbers. Concepts with which the reader may not already
be familiar are introduced and explained to t he degree necessary for use within
the text at the location where the concept is first used.
Students who enroll in ordinary differential equations courses normally do
so for only one or two semesters as an undergraduate. In addition, few of these
students ever enroll in a numerical analysis course. However, most students
who complete a differential equations course find employment in business,
industry, or government and will use a computer and numerical methods to
solve mathematical problems almost exclusively. Consequently, one objective
of this text is to solve ordinary differential equations in the same way they are
solved in many professions- by computer. Thus, the single most useful and
distinguishing feature of this text is the use of computer software through-
out the entire text to numerically solve various types of ordinary differential
equ ations. Prior to generating a numerical solution, appli cable theory must be
considered; therefore, we state (but usually do not prove) existence, unique-
ness, and continuation theorems for initial value problems at various points in
the text. Numerical case studies illustrate the possible pitfalls of comput ing
a numerical so lution without first considering the appropriate theory.
Differential equations are an important tool in constructing mathemati-
cal models for physical phenomena. Throughout the text, we show how to
numerically so lve many interesting mathematical models- su ch as popula-
t ion growth models, epidemic models, mixture problems, curves of pursuit,
the Richardson's arms race model, Lanchester's combat models, Volterra-
Lotka prey-predator models, pendulum problems, and the restricted three-
body problem. When feasible we develop models entirely within separate
sections. This gives the instructor more flexibility in selecting the material
to be covered in the course. We hope to enrich and enliven the study of
differential equations by including several biographical sketches and historical