The eigenvalue problems related to the second order linear self-adjoint differential equations which occur frequently in many fields of applied mathematics are
reviewed. Variational methods are applied to solve approximately the boundary value problem of this kind. The Rayleigh-Ritz method is utilized to approximate not
only the eigenvalues, but also the corresponding upper bounds for the eigenvalues of the eigenvalue problem. The basis set of polynomials, the basis set of
normalized polynomials and the basis set of the Gegenbauer polynomials are tested. Numerical experiments yield more accurate numerical results when a basis set of
the Gegenbauer polynomials is used. The Lehmann's method is also studied with the basis set of Gegenbauer polynomials to find the corresponding lower bounds for
the eigenvalues.

Keywords : The Rayleigh-Ritz Method, The Lehmann Method, The Basis Set of Gegenbauer Polynomials, Variational Methods.