Exact solutions of the Cauchy problem for the Friedmann equation

The cosmological Friedmann equation for the universe filled with a scalar field is reduced to a system of two equations of the first order, one of which is an equation with separable variables. For the second equation the exact solutions are given in closed form for potentials as constants and exponents. For the same equation exact solutions for quadratic potential are written in the form of a series in the attractor and spiral areas (inflation stage and the late-time acceleration of the universe respectively). Also exact solutions for very arbitrary potentials are given in the neighborhood of endpoint and infinity. The existence and uniqueness of classical solutions of the Cauchy problem for the Friedman equation in some cases and the presence of exactly two solutions in other cases is proved.