Consider a spring system whose equation is given by $$my''+\mu y'+ky=0$$ and let $D=\mu^2-4mk$. Now there are three cases and I am considering the cases that $D=0$ and $D>0$:
- When $D=0$, the solution is of the form $y=(a+bt)e^{rt}$. (Critically damped)
- When $D>0$, the solution is of the form $y=c_1e^{r_1t}+c_2e^{r_2t}$. (Overdamped)
While I understand that these two cases are very different and the function $y=(a+bt)e^{rt}$ is very different from the function $y=c_1e^{r_1t}+c_2e^{r_2t}$, it also seems to me that the two functions have very similar graphs (in particular, very similar end behaviours).
Question
- Why are the graphs of the solution in these two cases so similar (while in the other case $D<0$, the graph is very different)?
- How to tell from the graph whether we have a critically damped system or an overdamped system?