Euler's Method

The simplest method of solving ODE's is Euler's method:

y_n+1 = y_n + hf(y_n).

In order to integrate from the initial data at t = 0 up to t = T, divide the interval into N equal steps of size h = T/N and approximate y(t_n = nh) = y_n. This method works and is sometimes used in practice, but much better alternatives are described below. Nonetheless, it is the conceptual basis of all other methods, and a little analysis gives insight into how they all work.