On paradoxes of the formal approach to determining the eigenvalues of one-dimensional boundary value problems

The conventional approach to determining the eigenvalues of a one-dimensional boundary value problem consists in writing out the solution of the differential equation in general form containing indeterminate coefficients and constructing a system of homogeneous linear algebraic equations for these coefficients on the basis of the expressions for the boundary conditions. The eigenvalue is determined from the condition that the determinant of the system thus constructed is zero. In the classical problems (of string, rod, etc. vibrations), this method, as a rule, does not cause any difficulties, although several examples in which the zero value of the frequency satisfying the characteristic equation thus constructed is not an eigenfrequency were constructed and investigated, for example, in [1, p. 220]. We show that in some cases more complicated than the classical ones a similar situation can lead to paradoxical conclusions and erroneous results.