In this paper, we study the bending of a clamped thin orthotropic plate by the Bubnov–Galerkin method. Special-type polynomials constructed on the basis of classical Jacobi polynomials and satisfying the clamped conditions are used as basis functions. The “quasi-orthogonality” property of first and second derivatives of the polynomials allows us to immediately write out the formula for the deflection of the plate in the form of a series, by analogy with the Navier method for a free-supported isotropic plate. This solution approximates well the displacement of the plate, but the series for moments and shear forces do not give reliable results. The refusal of using the “quasi-orthogonality” property of the polynomials leads to the solution of an infinite system of linear algebraic equations for finding unknown coefficients of series. The resulting series give reliable results for both displacements and moments and shear forces. The convergence of the series for displacement is very good, but it worsens for shearing forces and bending moments.