In this study, the Dirichlet boundary problem for vibration of a parallelogram-shaped membrane is solved. The simplicity and transparency of the proposed procedures allow one to clarify the specific features of some state-of-the-art approaches to solve similar problems of mathematical physics. For many types of domains, including a wide range of non-canonical ones, the use of the concept of a general solution of the boundary value problem makes it possible to construct a numerical-analytical solution to the problem. In this case, sets of partial solutions for the basic equations of mathematical physics are used. The main idea is to indicate effective ways to determine arbitrary coefficients and functions that are part of a general solution. The conventional approach for deriving numerical-analytical solutions is used based on the mean square deviation minimization and collocation methods.