Variational simulation of the spectral problem

The ordinary fourth-order differential equation which is the zero approximation of the eigenvalue boundary problem is solved by the variational method to produce approximate formulas for eigenvalues. To obtain an explicit formula for eigenvalues, a transition is made from the differential problem to the variational problem in the Galerkin form. Calculating integrals in it gives a general formula for eigenvalues. The selection of functions satisfying certain boundary conditions yields approximate formulas suitable for the analysis of multiparameter dependencies. In particular, it is shown how the lowest eigenvalues are determined.