  We present a detailed study of the two Sun-like stars KIC 7985370 and KIC
7765135, aimed at determining their activity level, spot distribution, and
differential rotation. Both stars were discovered by us to be young stars and
were observed by the NASA Kepler mission. The stellar parameters (vsini,
spectral type, Teff, log g, and [Fe/H]) were derived from optical spectroscopy
which allowed us also to study the chromospheric activity from the emission in
the core of H\alpha\ and CaII IRT lines. The high-precision Kepler photometric
data spanning over 229 days were then fitted with a robust spot model. Model
selection and parameter estimation are performed in a Bayesian manner, using a
Markov chain Monte Carlo method. Both stars came out to be Sun-like with an age
of about 100-200 Myr, based on their lithium content and kinematics. Their
youth is confirmed by the high level of chromospheric activity, comparable to
that displayed by the early G-type stars in the Pleiades cluster. The flux
ratio of the CaII-IRT lines suggests that the cores of these lines are mainly
formed in optically-thick regions analogous to solar plages. The model of the
light curves requires at least seven enduring spots for KIC 7985370 and nine
spots for KIC 7765135 for a satisfactory fit. The assumption of longevity of
the star spots, whose area is allowed to evolve in time, is at the heart of our
approach. We found, for both stars, a rather high value of the equator-to-pole
differential rotation (d\Omega~0.18 rad/day) which is in contrast with the
predictions of some mean-field models of differential rotation for
fast-rotating stars. Our results are instead in agreement with previous works
on solar-type stars and with other models which predict a higher latitudinal
shear, increasing with equatorial angular velocity.
