  In the paper we describe a superexponentially convergent numerical-analytical
method for solving the eigenvalue problem for the some class of singular
differential operators with boundary conditions. The method (FD-method) was
firstly proposed by V. L. Makarov and successfully combines the benefits of
using the {\it coefficient approximation methods} (CAM) and the homotopy
approach. The sufficient conditions which provides convergence of the proposed
method are stated and rigorously substantiated. The algorithm for the software
implementation of the proposed method is described too. A lot of numerical
examples are included in the paper. The examples confirm the theoretical
conclusions. We also have made the comparison between the results obtained by
FD-method and results obtained by the powerful software package for solving
Sturm-Liouville problems --- SLEIGN2.
