I'm being asked this question and I don't quite understand how to do it.
I kinda know how the fourth-order Runge-Kutta method works.
$K_1 = hf(x_n,y_n)$
$K_2 = hf(x_n+\frac{h}{2},y_n+\frac{k_1}{2})$
$K_3 = hf(x_n+\frac{h}{2},y_n+\frac{k_2}{2})$
$K_4 = hf(x_n+\frac{h}{2},y_n+k_3)$
$y_{n+1} = y_n + \frac{k_1}{6}+ \frac{k_2}{3}+ \frac{k_3}{3}+ \frac{k_4}{6}$
But how do I expand this to seventh-order? Is there an equation for $K_n$ I can follow and an equation for $y_{n+1}$ which extends up to $k_7$?
I've seen some higher order Runge-Kutta examples online which uses a Butcher tableau but it all seems really complex to me.