The Euler Method

It is instructive to examine the simplest method, the Euler method, for solving the first-order scalar IVP given by

(This method is also called the Euler-Cauchy, forward Euler, or explicit Euler method).

In this method, the value of the dependent variable at the current point is calculated by straight line extrapolation from the previous point. Since the initial data is known, we can evaluate

and from this, we can calculate an approximation to by using the first two terms of a Taylor' series

We let , and define our approximation to as . Thus

Similarly,

and, in general, for

, where

(2.1)