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Brownian oscillator, i.e. a micron-sized or smaller particle trapped in athermally fluctuating environment is studied. The confining harmonic potentialcan move with a constant velocity. As distinct from the standard Langevintheory, the chaotic force driving the particle is correlated in time. Thedynamics of the particle is described by the generalized Langevin equation withthe inertial term, a coloured noise force, and a memory integral. We considertwo kinds of the memory in the system. The first one corresponds to theexponentially correlated noise in a weakly viscoelastic fluid and in the secondcase the memory naturally arises within the Navier-Stokes hydrodynamics. Exactanalytical solutions are obtained in both the cases using a simple andeffective method not applied so far in this kind of problems.

Brownian oscillators driven by correlated noise in a moving trap