I understand that the model problem $y'=\lambda y $ is always used to analyze the stability of a method. My question is why? Why this particular model problem? Why not do it for the question itself?
Example
$y'(t)=-2y(t)+1$, with $y(0)=1$.
I tried applying what was taught to me, analyzing the stability of the method Forward Euler, set $\frac{|y^{n+1}|}{|y^{n}|}\leq1$
to get $\frac{|(1-2k)y^n+k|}{|y^n|}$ and I am unable to proceed. But it seems that the model problem makes stability analysis easier. But does the result generalizes to this problem as well?