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 the potentials as constants and exponents. For the same equation, the exact solutions for a 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 the exact solutions for every arbitrary potential are given in the neighborhood of endpoint and infinity. The existence of all these classical solutions is proven.
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