An Euler-Lagrange-type equation is solved to different orders for aplane metallic system perturbed by the externa! operator Q used previously in a sum-rule calculation. This approach managed to reproduce to within a few percent the multipole surface-plasmon energies. The jellium model is used for the positive ionic background and local-density functionals for the Hamiltonian. From the zeroth-order Euler-Lagrange equation a self-consistent ground-state density is obtained and used in the first-order equation to obtain the induced density. From the induced density, several aspects of the system response are considered and special attention is focused on the static polarizability.