  Here we analyze the collective excitations as well as the expansion of a
trapped Bose-Einstein condensate with a vortex line at its center. To this end,
we propose a variational method where the variational parameters have to be
carefully chosen in order to produce reliable results. Our variational
calculations agree with numerical simulations of the Gross-Pitaevskii equation.
The system considered here turns out to exhibit four collective modes of which
only three can be observed at a time depending of the trap anisotropy. We also
demonstrate that these collective modes can be excited using well established
experimental methods such as modulation of the s-wave scattering length.
