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

y_{n+1} = y_{n} + hf(y_{n}).

(4)

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.