Forth order Runge-Kutta Method

The fourth order Runge-Kutta method could be summarized as:

Algorithm

$\dot y = f\left( {x,y} \right)$
$k_1 = hf\left( {x_n ,y_n } \right)$
$k_2 = hf\left( {x_n + {h \over 2},y_n + {{k_1 } \over 2}} \right)$
$k_3 = hf\left( {x_n + {h \over 2},y_n + {{k_2 } \over 2}} \right)$
$k_4 = hf\left( {x_n + h,y_n + k_3 } \right)$
$y_{n + 1} = y_n + {{k_1 } \over 6} + {{k_2 } \over 3} + {{k_3 } \over 3} + {{k_4 } \over 6}$