HCR - Mid Point Method

Formula (not the final one, read at the end):$x(t_0+h)=x(t_0)+h\dot{x}(t_0)+\frac{h^2}{2}\ddot{x}(t_0)$Where:

• $\dot{x}(t_0)=f(x(t),t)$
  ⇒ $x(t_0)=hf(x(t_0),t_0)+\frac{h^2}{2}\dot{f}(x(t_0),t_0)$

==I dont want to compute the derivative over time of $f(x(t_0),t_0)$==;
⇒ So we want the result of $\ddot{x}(t_0)$ using only $x(t_0)$ and $f(x(t),t)$, so we can:

Resulting in: Mid Point Method: $x(t_0+h)=x(t_0)+hf(x_0+\frac{h}{2}f(x_0))$