  We analyze solutions to a superconducting gap equation based on the
two-dimensional Hubbard model with nearest and next-to-nearest neighbor
hopping. The Cooper pair potential can be calculated exactly and expressed in
terms of elliptic functions. The Fermi surfaces at finite temperature and
chemical potential are calculated based on the exact two-body S-matrix of the
Hubbard model using the formalism we recently developed, which allows variation
of hole doping. The resulting solutions to the gap equation are strongly
anisotropic, namely largest in the anti-nodal direction, and zero in the nodal
directions of the Brillouin zone. For U/t = 13 and t' /t =-0.3, appropriate to
BSCO, and a physically natural choice for the cut-off, our self-contained
analytic calculations yield the gap in the anti-nodal direction Delta/t = 0.06
and a maximum Tc/t = 0.04 at hole doping h=0.15. For phenomenological fits to
the Fermi surfaces for cuprates, we obtain the comparable value Tc/t = 0.03 at
optimal doping, both in good agreement with experiments. The superconducting
gap is non-zero for all hole-doping h < 0.35, and increases all the way down to
zero doping, suggesting that it evolves into the pseudogap.
