Runge-Kutta (RK) methods are based on Taylor expansion formulae, but yield in general better algorithms for solutions of an ODE. The basic philosophy is that it provides an intermediate step in the computation of \( y_{i+1} \).

To see this, consider first the following definitions $$ \begin{equation} \frac{dy}{dt}=f(t,y), \tag{34} \end{equation} $$ and $$ \begin{equation} y(t)=\int f(t,y) dt, \tag{35} \end{equation} $$ and $$ \begin{equation} y_{i+1}=y_i+ \int_{t_i}^{t_{i+1}} f(t,y) dt. \tag{36} \end{equation} $$