The Bardeen-Cooper-Schrieffer (BCS) gap equation is solved analytically in one, two and three dimensions, for temperatures close to zero andTc. We work in the weak coupling limit, but allow the interaction widthν≡ħωm/EF to lie in the interval (0, ∞) Here,ħωm is the maximum energy of a force-mediating boson, andEF denotes the Fermi energy. We obtain expressions forTc and ΔC, the jump in the electronic specific heat acrossT=Tc, in the limitsν≪1 (the usual phonon pairing) andν>1 (non-phononic pairing). This enables us to see howTc scales with the mediating boson cut off. Our results predict a larger jump in the specific heat for the caseν>1, compared toν≪1. We also briefly touch upon the role of a van Hove singularity in the density of states.