  The bound state solution of the radial Schr\"{o}dinger equation with the
generalized Woods-Saxon potential is carefully examined by using the Pekeris
approximation for arbitrary $\ell$ states. The energy eigenvalues and the
corresponding eigenfunctions are analytically obtained for different $n$ and
$\ell$ quantum numbers. The obtained closed forms are applied to calculate the
single particle energy levels of neutron orbiting around $^{56}$Fe nucleus in
order to check consistency between the analytical and Gamow code results. The
analytical results are in good agreement with the results obtained by Gamow
code for $\ell=0$.
