When solving homogeneous second order constant coefficient linear ODEs ($ay''+by'+cy=0$), there are three 'cases' that solutions fall into, based on the roots $r\_1$, $r\_2$ of the characteristic equation $ar^2+br+c=0$:

* $r\_1, r\_2 \in \Bbb R,\; r\_1 \ne r\_2$, aka "overdamped", for which the general solution is of the form $y(t) = c\_1 e^{r\_1 t} + c\_2 e^{r\_2 t}$.
* $r\_1, r\_2 \in \Bbb C,\; r\_1=\overline{r\_2}$, aka "underdamped", for which the general solution is $y(t) = c\_1 e^{\Re(r)\,t}
 \cos\Im(r)\,t + c\_2 e^{\Re(r)\,t} \sin\Im(r)\,t$.
* $r\_1, r\_2 \in \Bbb R,\; r\_1 = r\_2$, aka "critically damped", for which the general solution is $y(t) = c\_1 e^{rt} + c\_2 t e^{rt}$.

I understand the underdamped solution as essentially the same thing as the overdamped solution, equivalent to picking complex $c\_1, c\_2$ such that $y(t)$ is always real. But the critically damped case doesn't seem to fit neatly as a generalization of the other cases - that extra 't' makes it at least appear like it's something else entirely.

Where does the "$t$" actually come from in the critically damped case?
Why don't the other cases require or allow this?