  The formulation of Brans-Dicke (BD) gravity with matter in the Einstein frame
is realized as Einstein gravity with dilaton and dilaton coupled matter. We
calculate the one-loop 4d anomaly-induced effective action due to N dilaton-
coupled massless fermions on the time-dependent conformally flat background
with non-trivial dilaton. Using that complete effective action the
(fourth-order) quantum corrected equations of motion are derived. One special
solution of these equations representing an inflationary Universe (with
exponential scale factor) and (much slower) expanding BD dilaton is given.
Similarly, the 2d quantum BD Universe with time-dependent dilaton is
constructed. In the last case the dynamics is completely due to quantum
effects.
