Differential Equations and Linear Algebra, 4th Edition

2.5 A Closer Look at the Euler Method

The Euler method as presented in Section 2.4 is not often used in practice, mainly because more accurate methods are available. But Euler’s method has the advantage of simplicity, and a careful study of this method yields insights into the workings of more accurate methods, because many of the latter are extensions or refinements of the Euler method.

To compare two different methods of numerical approximation, we need some way to measure the accuracy of each. Theorem 1 tells what degree of accuracy we can expect when we use Euler’s method.

THEOREM 1 The Error in the Euler Method

Suppose that the initial value problem

$${\displaystyle \frac{dy}{dx}}=f(x,y),y(x_0)=y_0$$ (1)

has a unique solution y(x) on the closed interval [a, b] with ...