A novel method for obtaining the distribution functions from the Boltzmann equations for binary granular gas mixtures of smooth spheres has been developed. The method is capable of producing accurate results irrespective of the values of the parameters which define the system. Here we explain the method and present results obtained using it for the temperature ratio of the components, as well as for the flatness (and higher order measures of flatness) of the distribution functions, for the homogeneous cooling state. It turns out that the mean field approximation for the temperature ratio yields results which are within about 1 or better of the exact results for all checked values of the parameters (except when the mass ratio is very large (or small) and the system is very inelastic) even when the values of the flatness suggest that the distribution functions are not near-Maxwellian. The use of the method for obtaining constitutive relations is outlined but detailed results are deferred to another publication.
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