We solve the Eliashberg equations for the case of an explicit dependence of the interactions, and of the resulting self-energies XI (I?, a), X2(% w ) . We consider a strong energy-dependence of the electron-electron scattering-rate r;', which is associated with a strong energy-dependence of the electron-phonon matrix element g(k,k'). We characterize this energy-dependence by a cutoff tl, which is of the order of the phonon frequency Up),. We find that we can account for a large number of unexpected features of the superconductivity of the cuprates by the BCS electron-phonon theory, if we consider very large values of the McMillan coupling constant Aph, and small values of the cutoff C1. Specifically, the Coulomb interaction is found not to depress T,; thejsotope effect is strongly reduced when c1