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In the paper we investigate well-defined solvability of the Dirichlet problem for a second order partial operator-differential equation in a Hilbert space. First we prove a theorem on the isomorphism of the principle part of the given equation. Then it is proved that operator coefficients of the disturbed part of the equation can be chosen from a wider class so that theorems on solvability of a boundary value problem for complete equations hold and are easily verifiable in practical problems. This feature strongly distinguishes our research from the works, where the solvability conditions entail arbitrary smallness of the disturbed part of equations and they are expressed by means of restrictions on the resolvent growth of the appropriate operator beam. [ABSTRACT FROM AUTHOR]