ABSTRACT
In this chapter, the authors discuss in detail the first-order initial value problem. They define the direction field for the differential equation, discuss the significance of the direction field, and show how to use a computer program to produce a graph of the direction field. The authors state a fundamental existence theorem, a fundamental existence and uniqueness theorem, and a continuation theorem for the initial value problem. They show how to apply these theorems to a variety of initial value problems and illustrate and emphasize the importance of these theorems. The authors discuss how to obtain the general solution 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. They present some simple numerical techniques for solving first-order initial value problems.
