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 Halpha 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 (dOmega~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.
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