On setting (3.22) differs from the expansion (3.16) for by a term of order h. Thus, the resulting method, which is the Euler's method, has order one.

Now let , so that and (3.21) reduces to

(3.23)

Matching this with the expansion (3.16), we have

(3.24)

This gives a set of two equations in three unknowns and there exists a one parameter family of solutions. Thus there exists an infinite number of two–stage Runge–kutta methods of order two and none of order more than two. Two particular solutions of (3.24) yield the following well known methods:

i) .

The resulting method is

(3.25)