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?