  A generalized dynamical equation for the scale factor of the universe is
proposed to describe the cosmological evolution, of which the $\Lambda$CDM
model is a special case. It also provides a general example to show the
equivalence of the modified equation of state (EOS) and a scalar field model.
In the mathematical aspect, the EOS, the scalar field potential $V(\phi)$, and
the scale factor $a(t)$ all have possessed analytical solutions. Such features
are due to a simple form invariance of the equation inherited which determines
the Hubble parameter. From the physical point of view, this dynamical equation
can be regarded as the $\Lambda$CDM model with bulk viscosity, an existence
content in the universe. We employ the SNe Ia data with the parameter
$\mathcal{A}$ measured from the SDSS data and the shift parameter $\mathcal{R}$
measured from WMAP data to constrain the parameters in our model. The result is
that the contribution of the bulk viscosity, accumulated as an effective dark
energy responsible for the current cosmic accelerating expansion, is made
approximately ten percent to that of the cosmological constant.
