# DE: Critical Damping Oscillating?

I'm having a problem understanding a critical damping model. I know critical damping is supposed to return the system to equilibrium as quickly as possible without oscillating, and a critically damped system will have repeated roots so the general solution will be: c1e^rt + c2te^rt

But what happens when c2 is negative, for instance when solving y''+6y'+9y=0, y(pi)=-(pi-2)e^(-3pi), y'(pi)=(3pi-7)e^(-3pi), the characteristic equation is r^2 +6r +9, r =-3 repeated, then solving for the initial conditions I get: c1=2, c2=-1, then:

y(t)=2e^(-3t) -te^(-3t)

This function drops quickly to 0 at t=2, but then it crosses the origin. I thought it wasn't supposed to oscillate since it's critically damped? I've never taken a physics class so I think I must be missing some physical intuition or something here. Any help is appreciated, thanks.

## Answers and Replies

vela
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You're not missing anything. Oscillation means it will cross the t-axis repeatedly. If the system crosses just once, it's not oscillating.