TABLE OF CONTENTS We are concerned in the first instance with the computation of the solution of a first-order differential equation with one independent variable and a given initial condition, sometimes known as the initial value problem:
The general idea, given that we know the value of y at x = x 0, is to compute its value at x = x 0 + h, where h is sufficiently small. In first approximation, also known as Euler's approximation, this value may be given by the first terms of the Taylor expansion of y( x) about x 0 and is:
Once this value is known, the approximate value at x = x 0 + 2 h may be computed, and so on. We shall be interested in whether the values of y so calculated increasingly depart from their true values the larger is x ? x 0. We shall also need to use higher order terms in the Taylor expansion in order to obtain greater accuracy. Clearly, we must distinguish between the local, or truncation, error of such a formula, and the global, or accumulated, error after many steps.
First, however, we need to know in what circumstances a differential equation has a solution at all, and if it does whether it is unique. We also have to examine its stability. In the next chapter we shall extend the theory to systems of differential equations with...
