Para \( s = 2 \) $$ \begin{align*} k_{1} &= f(t_{n}, y_{n}) \\ k_{2} &= f(t_{n} + c_{2} h, y_{n} + h(a_{21} k_{1})) \\ y_{n+1} &= y_{n} + h (b_{1} k_{1} + b_{2} k_{2}) \end{align*} $$
tabla parametrizada de Butcher
\begin{array}{c|c c} 0 & & \\ \alpha & \alpha & \\ \hline & 1 - \frac{1}{2 \alpha} & \frac{1}{2 \alpha} \end{array}
si \( \alpha = \frac{1}{2} \)
\begin{array}{c|c c} 0 & & \\ \frac{1}{2} & \frac{1}{2} & \\ \hline & 0 & 1 \end{array}
reemplazando se obtiene el método del punto medio $$ \begin{align*} k_{1} &= f(t_{n}, y_{n}) \\ k_{2} &= f \left( t_{n} + \frac{1}{2} h, y_{n} + h \left( \frac{1}{2} k_{1} \right) \right) \\ y_{n+1} &= y_{n} + h k_{2} \end{align*} $$
si \( \alpha = \frac{2}{3} \)
\begin{array}{c|c c} 0 & & \\ \frac{2}{3} & \frac{2}{3} & \\ \hline & \frac{1}{4} & \frac{3}{4} \end{array}
reemplazando se obtiene el método de Ralston $$ \begin{align*} k_{1} &= f(t_{n}, y_{n}) \\ k_{2} &= f \left( t_{n} + \frac{2}{3} h, y_{n} + h \left( \frac{2}{3} k_{1} \right) \right) \\ y_{n+1} &= y_{n} + h \left( \frac{1}{4} k_{1} + \frac{3}{4} k_{2} \right) \end{align*} $$
si \( \alpha = 1 \)
\begin{array}{c|c c} 0 & & \\ 1 & 1 & \\ \hline & \frac{1}{2} & \frac{1}{2} \end{array}
reemplazando se obtiene el método de Heun $$ \begin{align*} k_{1} &= f(t_{n}, y_{n}) \\ k_{2} &= f( t_{n} + h, y_{n} + h k_{1}) \\ y_{n+1} &= y_{n} + h \left( \frac{1}{2} k_{1} + \frac{1}{2} k_{2} \right) \end{align*} $$