  We investigate a disordered multi-dimensional linear system in which the
interaction parameters vary stochastically in time with defined temporal
correlations. We refer to this type of disorder as "annealed", in contrast to
quenched disorder in which couplings are fixed in time. We extend Dynamical
Mean-Field Theory to accommodate annealed disorder and employ it to find the
exact solution of the linear model in the limit of a large number of degrees of
freedom. Our analysis yields analytical results for the non-stationary
auto-correlation, the stationary variance, the power spectral density, and the
phase diagram of the model. Interestingly, some unexpected features emerge upon
changing the correlation time of the interactions. The stationary variance of
the system and the critical variance of the disorder are generally found to be
a non-monotonic function of the correlation time of the interactions. We also
find that in some cases a re-entrant phase transition takes place when this
correlation time is varied.
