Show that the explicit Runge-Kutta scheme
\begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})]
\end{equation}
where $k_{1} = f(t,y_{n})$
applied to the equation $y'= y(1-y)$ has two spurious fixed points if $h>2$.
Briefy describe how you would investigate their stability.
=> my attempt so far
from $y'= y(1-y)$
$y'= 0$
$y=0$ or
$y=1$ which are the true fixed points.
after that i rearranged the runge kutta scheme
\begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})]
\end{equation}
\begin{equation} y_{n+1} = y_{n} + \frac{h}{2} [f(t,y_{n} + f(t+h, y_{n}+hf(t,y_{n})]
\end{equation}
i try to put the fixed points into above scheme and try to get two two Spurious fixed point for $y_{n}$ but i got struck. for the stability to describe i need to get two Spurious fixed point first. but in general please help to describe stability too because this part i really get confuse. Anyone please help me, it will be really helpful for my other problems too if i got this answer correctly.