  A semiclassical second-order differential equation for the inhomogeneous
local gap $\Delta(r)$ is derived from a strict second-order $\hbar$ expansion
of the anomalous pairing tensor and compared with a similar equation given by
Simonucci et al. The second-order normal density matrix is given as well.
Several extra gradient terms are revealed. Second-order expressions at finite
temperature are given for the first time. The corresponding Ginzburg-Landau
equation is presented and it is shown that, compared to the equation of Baranov
and Petrov, an extra second-order gradient term is present. Applications to the
pairing gap in cold atoms in a harmonic trap are presented.
