The task of solving a two-point problem for a system of ordinary differential equations with variable coefficients arises when analyzing a contact problem of elasticity theory using the integral transformation method. The effective method for constructing such a solution in Cartesian coordinates is known from the papers by V.M. Alexandrov and S.M. Aizikovich. The present paper presents a unified method for reducing the contact problems of elasticity theory for continuously inhomogeneous half-bounded media to the solution of dual integral equations generalized for the non-Cartesian coordinates. The inhomogeneity of the medium is modeled by the assumption that elastic moduli are arbitrary smooth enough functions of the acting loading coordinate. Practical implementation of the method revealed its high efficiency both for small and large values of the characteristic geometric parameter of the medium and made it possible to investigate in detail a wide class of problems for wedge, cylindrical and spherical regions with the elastic moduli varying along angular or radial coordinates.
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