  We obtain general solutions for some flat Friedmann universes filled with a
scalar field in induced gravity models and models including the
Hilbert-Einstein curvature term plus a scalar field conformally coupled to
gravity. As is well known, these models are connected to minimally coupled
models through the combination of a conformal transformation and a
transformation of the scalar field. The explicit forms of the self-interaction
potentials for six exactly solvable models are presented here. We obtain the
general solution for one of the integrable models, namely, the induced gravity
model with a power-law potential for the self-interaction of the scalar field.
We argue that although being mathematically in a one-to-one correspondence with
the solutions in the minimally coupled models, the solutions in the
corresponding non-minimally coupled models are physically different. This is
because the cosmological evolutions seen by an internal observer connected with
the cosmic time can be quite different. The study of a few induced gravity
models with particular potentials gives us an explicit example of such a
difference.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;Exactly Solvable and Integrable Systems