In a weighted Sobolev space for a non-homogeneous boundary value problem of a higher-order operator-differential equation, the well-posedness and the unique solvability is established. The estimates on the norms of the operators of intermediate derivative associated with the solvability conditions are provided. In the principal part of the differential equation, the relation between the lower bound of the spectrum of the main operator and the weight exponent 𝑘 (−∞ < 𝑘 < ∞) is determined. The results are illustrated as an applied example of a problem for partial differential equations