I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: $e^{-\alpha t}(Acos(\omega_d t) + Bsin(\omega_d t))$

I am interested in a solution of the form $e^{-\alpha t^2}Acos(\omega_d t)$ i.e., I want the oscillations to die at quadratic rate.

Is there a corresponding differential equation that can generate this kind of behavior?

Note: I asked this question before. The answer I got was a trivially constructed linear time-varying system. I am interested in a more compact and physically driven representation.