2nd order Runge-Kutta method for linear ODE Could someone please help me with the next step of this 2nd order Runge-Kutta method?
I am solving the following initial value problem (IVP)
$$x' = - \frac12 x(t), \qquad x(0)=2$$
I wish to use the second order Runge-Kutta method
$$x(t+h)=x(t)+1/2(F_1+F_2),$$
where \begin{align*}
F_1&=hf(t,x) \\
F_2&=hf(t+h,x+F_1).
\end{align*}
Let us use a spacing of $h=1$.
My working goes like this:
$$F_1 = -\frac{x(0)}{2}=-1.$$
Then
\begin{align*}F_2&=1\times f(0+1,x(0)+F_1) \\
&=-1/2 \times x(1).\end{align*}
But I have no idea what $x(1)$ is.

$F_1$ is a constant. $F_2$ is a function of $x_1$. So on the right hand side of the original equation you have a $x_1$ term as well as the LHS is itself $x_1$ Bring them to the same side and solve
