  We show that the width of the wave-packet of a class of generalized nonlinear
Schrodinger equations (NLSE) trapped in an arbitrary time-dependent harmonic
well in any dimensions is universally determined by the same Hill's equation.
This class of generalized NLSE is characterized by a dynamical O(2,1) symmetry
in absence of the trap. As an application, we study the dynamical instabilities
of the rotating as well as non-rotating Bose-Einstein condensates in one and
two dimensions. We also show exact extended parametric resonance in a
non-relativistic Chern-Simons theory producing a gauged NLSE.
