  We present a complete analytical resolution of the one dimensional Burgers
equation with the elastic forcing term $-\kappa^{2} x+f(t)$,
$\kappa\in\mathbb{R}$. Two methods existing for the case $\kappa=0$ are adapted
and generalized using variable and function transformations, valid for all
values of space an time. The emergence of a Fokker-Planck equation in the
method allows to connect a fluid model, depicted by the Burgers equation, with
an Ornstein-Uhlenbeck process.
