2nd order Runge-Kutta

Euler's method rests on the idea that the slope at one point can be used to extrapolate to the next. A plausible idea to make a better estimate of the slope is to extrapolate to a point halfway across the interval, and then to use the derivative at this point to extrapolate across the whole interval. Thus,

$$k=\Delta x f(x_n,y_x),$$ (12)

$$y_{n+1}=y_n+\Delta x f(x+1/2\Delta x, y_n+1/2k) + O(\Delta x^3).$$ (13)

It has the same accuracy as the Taylor series (5). It requires the evaluation of $f$ twice for each step.