To solve the kinetic equation for particles of a monodisperse two-phase mixture the method of successive approximations is developed; this resembles in its main features the well-known Chapman-Enskog method in the kinetic theory of gases. This method is applicable for a mixture whose state differs slightly from the equilibrium, i.e., when time and space derivatives of the dynamic variables describing the mean flow of both phases of the mixture are sufficiently small. Accordingly, the solution obtained is valid when the time and space scales of the mean flow exceed considerably those for random pseudo-turbulent motion of particles and a fluid. The conservation equations for determination of all the dynamic variables are formulated in approximations which have the same meaning as those of Euler and Navier & Stokes in hydromechanics of one-phase media.