Chapter 2
The Initial Value Problem
y' = f(x, y); y(c) = d
In this chapter we discuss in detail the first-order initial value problemy' = f(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 obtain the general so lution to first-order differential equations
which are separable or linear and, thereby, solve initial value problems in which
the differential equation is separable or linear. Next, we present some simple
numerical techniques for solving first-order initial value problems. Finally, we
explain how to use a computer program to generate approximate, numerical
solutions to first-order initial value problems. We illustrate and interpret the
various kinds of results which computer software may produce. 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.
DEFINITION First-Order Initial Value ProblemThe first-order initial value problem is to solve the differential equa-
tion (DE)(1) y' = f(x, y)
subject to the constraint, called an initial condition (IC),(2) y(c) = d.
It is customary to write this initial value problem (IVP) more compactly as
(3) y' = f(x, y); y(c) = d.
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