Restoration of the linear potential in the sturm-liouville problem

The problem of identifying the variable coefficient of elasticity of a medium with respect to natural frequencies of a string oscillating in this medium is considered. A method for solving the problem is found, based on the representation of linearly independent solutions of the differential equation in the form of Taylor series with respect to two variables, substituting them into the frequency equation, and determining the unknown coefficients of the linear function from this frequency equation. An analytical method has also been developed that allows one to prove the uniqueness or nonuniqueness of the restored polynomial coefficient of elasticity of a medium by a finite number of natural frequencies of oscillations of a string, and also to find a class of isospectral problems, that is, boundary value problems for which the eigenvalue spectra coincide. The latter is based on the method of variation of an arbitrary constant. We consider examples of finding isospectral classes, and also unique boundary value problems having a given spectrum.