Note: It is of interest to note that this is precisely the number of undetermined coefficients we had at our disposal when we derived the general three-stage explicit Runge-Kutta method. There, however, the form of the expansion for precluded any possibility of attaining an order greater than three. In the present case, the form of the expansion for holds out a possibility of attaining order four.

We now have eight equations to be satisfied but have only six coefficients at our disposal. However, if we solve (6.8), (6.9) and the last equations of (6.10) and (6.11), we find

(6.12)

with superior alternative signs to be taken together. On substituting these values into the rest of equations, we find that all of the remaining equations of (6.8)-(6.11) are satisfied if