Page 193 - Schaum's Outline of Differential Equations

CHAPTER 19

Further Numerical

Methods for Solving

First-Order Differential

Equations

GENERAL REMARKS
As we have seen in the previous chapter, graphical and numerical methods can be very helpful in obtaining
approximate solutions to initial-value problems at particular points. It is interesting to note that often the only
required operations are addition, subtraction, multiplication, division and functional evaluations.
In this chapter, we consider only first-order initial-value problems of the form

Generalizations to higher-order problems are given in Chapter 20. Each numerical method will produce
approximate solutions at the points x 0, x^, x 2, ..., where the difference between any two successive.x-values is
a constant step-size h; that is, x n +1 - x n = h (n = 0, 1,2,...). Remarks made in Chapter 18 on the step-size
remain valid for all the numerical methods presented below.
The approximate solution at x n will be designated by y(x n), or simply y n. The true solution at x n will be
denoted by either Y(x n) or Y n. Note that once y n is known, Eq. (19.1) can be used to obtain y' n as

The simplest numerical method is Euler's method, described in Chapter 18.
A predictor-corrector method is a set of two equations for y n +1. The first equation, called the predictor, is
used to predict (obtain a first approximation to) y n + \, the second equation, called the corrector, is then used to
obtain a corrected value (second approximation) to y n + 1. In general, the corrector depends on the predicted
value.

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