If exponentials are eigenfunctions, where does the transient term come from?
There is a transient because the input isn't just a cosine, but a causal cosine; that is, the input isn't a cosine from $-\infty \rightarrow \infty$, only from 0 onwards.  If the input were actually just an infinite length cosine, then you would not get any transients.

I notice that the notes you are working from don't show that the input is causal.  One reason I maintain that it actually is, is that otherwise you have an output that is tending to $\infty$ as $t \rightarrow -\infty$.  And also you don't get transients without a change to the system; the change in this case is that the system starts at time $t=0$.