This is the definition of numerical stability I have learned:
"A numerical method $ u_n $ is stable if $ |u_n|\to 0 $ as $ n \to \infty $ ".
It is easy to show numerical stability if the IVP to show for takes the form of
$ y' = \lambda y \\ y(0) = y_0 $
With the forward Euler method you get
$ y_{n+1} = y_n(1 + h\lambda)$
and if $ |1 + h\lambda|<1$ then $ y_{n+1} < y_n $ and hence it is stable.
However, if I am given the IVP
$ y' = \lambda y + g(t) \\ y(0) = y_0 $
then it becomes not so easy anymore. I get
$ y_{n+1} = y_n(1 + h\lambda) + hg_n(t)$
and I don't know how to proceed from here, since I can't factorize g(t). However, it is my understanding that the region of stability is still given by
$ |1 + h\lambda|<1 $
even though we have a non-homogeneous ODE. Can someone show me how I can find the stability region of the non-homogeneous IVP.
