The ellipsoidally symmetric extension of Buda-Lund hydrodynamic model is shown here to yield a natural description of the long-unsolved problem of the pseudorapidity dependence of the elliptic flow, $v_2(\eta)$, as determined recently by the PHOBOS experiment for Au+Au collisions at $\sqrt{s_{NN}} = 19.7$, $\sqrt{s_{NN}} = 62.4$, $\sqrt{s_{NN}} = 130$ and $200$ GeV. There are essentially two parameters that describe the measured data. One is the expansion rate in the beam direction that corresponds to the width of the pseudorapidity distribution. The other is the momentum-space eccentricity that corresponds to the coordinate-space eccentricity under certain conditions. Further parameters are fixed to values obtained from fits to RHIC data on the p_t dependent elliptic flow, spectra and HBT radii. The description of the rapidity dependence of the elliptic flow is also in consonance with a natural resolution of the "RHIC HBT puzzle" within the framework of the Buda-Lund model.