The Numerical Solution of Ordinary Differential Equations

In this chapter we shall study the following three central problems of the computational theory for ordinary differential equations: Problem 1. Consider the ordinary differential equation $$y'(x)\, = f(x,y(x)),\,y({x_0})\, = \,{y_0},$$ (8.1) where x is a scalar variable, and y and f are N-tuple-valued functions of the indicated variables. The problem of finding the function y(x) satisfying (8.1) for a given f and y 0 is called an initial-value problem. The solution of equation (8.1) has to be determined in an interval [x 0, b].