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We develop a method for constructing exact cosmological solutions of theEinstein equations based on representing them as a second-order lineardifferential equation. In particular, the method allows using an arbitraryknown solution to construct a more general solution parameterized by a set of3\textit{N} constants, where \textit{N} is an arbitrary natural number. Thelarge number of free parameters may prove useful for constructing a theoreticalmodel that agrees satisfactorily with the results of astronomical observations.Cosmological solutions on the Randall-Sundrum brane have similar properties. Weshow that three-parameter solutions in the general case already exhibitinflationary regimes. In contrast to previously studied two-parametersolutions, these three-parameter solutions can describe an exit from inflationwithout a fine tuning of the parameters and also several consecutiveinflationary regimes.

The linearization method and new classes of exact solutions in cosmology