"Let $a\in \mathbb{R}$, consider the PVI: $u'(t)=u_0e^{at}(a \cos(t) − \sin(t))$, $t > 0$ $u (0) = u_0$ with $u (t)=u_0e^{at}\cos(t)$ being the exact solution. Study the conditioning with respect to the initial condition. Show that the problem is well conditioned if $a < 0$ and ill-conditioned if $a > 0$."
I'm having difficulty resolving this issue, in my understanding, to calculate the numerical conditioning $K(d)$, I would do $K(d)≈\dfrac{\Vert d \Vert}{\Vert G(d)\Vert}\Vert G '(d) \Vert$. In the question statement it is suggested using the definition of absolute conditioning number $K_{abs}(a)$. What should I do?