The backwards Euler method (implicit Euler scheme) is a numerical method for the finding the solution of ordinary differential equations, which is defined as follows, $$ y(t_{n+1}) \approx y(t_n) + hf(t_{n+1}, y(t_{n+1}))$$ where $h = t_{n+1} - t_n$.
How could one find and solve the difference equation resulting from using $f(y) =-y \,$ for this method? What does this then say about the unconditional stability of the backward Euler method?
We have $$y(t_{n+1})=y(t_n)-hy(t_{n+1})$$
Hence,
$$(1+h)y(t_{n+1})=y(t_n)$$
$$y(t_{n+1}) = \frac1{1+h}y(t_n)$$
$$y(t_{n})=\left(\frac1{1+h} \right)^n y(t_0)$$
Note that we have $0<\frac1{1+h}<1$ for $h>0$, it is unconditionally stable.