  We investigate the spin dynamics in the two-dimensional spin-orbit coupled
system subject to an in-plane ($x$-$y$ plane) constant electric field, which is
assumed to be turned on at the moment $t=0$. The equation of spin precession in
linear response to the switch-on of the electric field is derived in terms of
Heisenberg's equation by the perturbation method up to the first order of the
electric field. The dissipative effect, which is responsible for bringing the
dynamical response to an asymptotic result, is phenomenologically implemented
\`{a} la the Landau-Lifshitz-Gilbert equation by introducing damping terms upon
the equation of spin dynamics. Mediated by the dissipative effect, the
resulting spin dynamics asymptotes to a stationary state, where the spin and
the momentum-dependent effective magnetic field are aligned again and have
nonzero components in the out-of-plane ($z$) direction. In the linear response
regime, the asymptotic response obtained by the dynamical treatment is in full
agreement with the stationary response as calculated in the Kubo formula, which
is a time-independent approach treating the applied electric field as
completely time-independent. Our method provides a new perspective on the
connection between the dynamical and stationary responses.
