  We investigate a spherical overdensity model for the non-clustering dark
energy (DE) with the constant equation of state, w in a flat universe. In this
case, the exact solution for the evolution of the scale factor is obtained for
general w. We also obtain the exact (when w = - 1/3) and the approximate (when
w neq -1/3) solutions for the ratio of the overdensity radius to its value at
the turnaround epoch (y) for general cosmological parameters. Also the exact
and approximate solutions of the overdensity at the turnaround epoch (zeta) are
obtained for general w. Thus, we are able to obtain the non-linear overdensity
Delta = 1 + delta at any epoch for the given DE model. The non-linear
overdensity at the virial epoch (Delta_{vir}) is obtained by using the virial
theorem and the energy conservation. The non-linear overdensity of every DE
model converges to that of the Einstein de Sitter universe ~ 147 when
z_{vir}increases. We find that the observed quantities at high redshifts are
insensitive to the different w models. The low-redshift cluster (z_{vir} ~
0.04, i.e., z_{ta} ~ 0.7) shows the most model dependent feature and it should
be a suitable object for testing DE models. Also as Omo increases, the model
dependence of the observed quantities decreases. The error in the approximate
solutions is at most 2% for a wide range of the parameter space. Even though
the analytic forms of y and \zeta are obtained for the constant w, they can be
generalized to the slowly varying w. Thus, these analytic forms of the scale
factor, y, and zeta provide a very accurate and useful tool for measuring the
properties of DE.
