Here you can see that the transfer function applied to a cosine input will give you a sinusoid and a transient term:
$$ x(t) = \underbrace{(x(0) + x'(0))(2 e^{-t} - e^{-2t}) + \frac{2}{5} e^{-t} - \frac{1}{2}e^{-t}}_{{\rm goes\ to\ } 0 {\rm\ as\ } t \rightarrow \infty }\ \ \ + \frac{1}{10} \cos(t) + \frac{3}{10} \sin(t) $$
However, I don't understand how this can be, aren't exponentials eigenfunctions of LTI systems? so how come it can give an extra (transient) term?
At the same time it makes sense that it has a transient term from the CF of the function. How do I reconcile this?