if you fell into a tube through the earth

[The_Ryan]

After thinking it about it a bit more, I see now that my previous statement can be interpreted in such a way that it is incorrect. So let me amend it to this: there is no nontrivial solution to the CDHO case with an initial zero veloctiy in which an intersection with the equilibrium point lies in the future. I there a conter example I missed?

PS While most solutions to the CDHO cross once, the purely exponential solution won’t cross at all.

[Chronos]

D’oh! I was forgetting the zero initial velocity part. And there are certainly at least some cases of zero initial velocity, critically damped, where you never cross the axis: Take the one-crossing solution, and set t=0 at the turnaround point on the far side.