So we have the equation $$y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}hf(y_n+2) $$ where $h>0$ and $f(y)=-\lambda y$ , and $ \lambda >0 $.
Can someone just write down their general solution to this as I have one and I'm meant to show that the solution is unconditionally A-stable by showing that $\lim_{n\to \infty }(y_n)=0$ but i'm having trouble doing this with my general solution.
We can assume that the constants in the general solution are both positive and we pick $y_0 $ and $y_1$ such that $0<y_1<y_0$.