ABSTRACT
In this chapter we discuss in detail the first-order initial value problem y′ = 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 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. 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.
