The present paper is devoted to development of the homogeneous-solution method widely used to solve biharmonic boundary-value problems of elasticity theory, and to its application to a broader class of boundary-value problems with a self-adjoint operator of order 2n in a rectangular region. The forms of the eigenfunctions are established, the problem of constructing biorthogonality relations is solved, and a solution is given for the problem of determining the coefficients in n-fold expansions of a special type in closed form using the biorthogonality relations.