The numerical solution of the problem of guided propagation of polarized light in a smooth junction of a planar waveguide is considered. Within the framework of the model of adiabatic guided modes, the system of Maxwell equations is reduced to a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zeroth approximation and the same number of equations in the first approximation. The multilayer structure of waveguides makes it possible to reduce the problem to a homogeneous system of linear algebraic equations, whose nontrivial solvability condition yields the dispersion equation. Auxiliary eigenvalue problems for describing the adiabatic modes of the waveguide are solved.