  Brownian oscillator, i.e. a micron-sized or smaller particle trapped in a
thermally fluctuating environment is studied. The confining harmonic potential
can move with a constant velocity. As distinct from the standard Langevin
theory, the chaotic force driving the particle is correlated in time. The
dynamics of the particle is described by the generalized Langevin equation with
the inertial term, a coloured noise force, and a memory integral. We consider
two kinds of the memory in the system. The first one corresponds to the
exponentially correlated noise in a weakly viscoelastic fluid and in the second
case the memory naturally arises within the Navier-Stokes hydrodynamics. Exact
analytical solutions are obtained in both the cases using a simple and
effective method not applied so far in this kind of problems.
