To demonstrate the philosophy behind RK methods, let us consider the second-order RK method, RK2. The first approximation consists in Taylor expanding \( f(t,y) \) around the center of the integration interval \( t_i \) to \( t_{i+1} \), that is, at \( t_i+h/2 \), \( h \) being the step. Using the midpoint formula for an integral, defining \( y(t_i+h/2) = y_{i+1/2} \) and \( t_i+h/2 = t_{i+1/2} \), we obtain $$ \begin{equation} \int_{t_i}^{t_{i+1}} f(t,y) dt \approx hf(t_{i+1/2},y_{i+1/2}) +O(h^3). \tag{37} \end{equation} $$ This means in turn that we have $$ \begin{equation} y_{i+1}=y_i + hf(t_{i+1/2},y_{i+1/2}) +O(h^3). \tag{38} \end{equation} $$