The family of explicit Runge–Kutta methods is given by $$ y_{n+1}=y_n+\sum_{i=1}^{s}{b_i k_i}, $$ where $$\begin{align} k_1 & =hf(t_n,y_n),\\ k_2 & =hf(t_n+c_2 h, y_n+a_{21}k_1),\\ k_3 & =hf(t_n+c_3 h, y_n+a_{31}k_1+a_{32}k_2),\\ \vdots\\ k_3 & =hf(t_n+c_s h, y_n+a_{s1}k_1+a_{s2}k_2+a_{s3}k_3+\cdots+a_{s,s-1}k_{s-1}) \end{align}$$
But how Runge and Kutta achieved it? Where this formula come from and what are $k_i\text{'s}$?
