  In this paper, the approximate analitical solutions of the hyper-radial
Schr\"{o}dinger equation are obtained for the generalized Wood-Saxon potential
by implementing the Pekeris approximation to surmount the centrifugal term. The
energy eigenvalues and corresponding hyper-radial wave functions are found for
any angular momentum case via the Nikiforov-Uvarov (NU) and Supersymmetric
quantum mechanics (SUSY QM) methods. Hence, the same expressions are obtained
for the energy eigenvalues, and the expression of hyper-radial wave functions
transformed each other is shown owing to these methods. Furthermore, a finite
number energy spectrum depending on the depths of the potential well $V_{0}$
and $W$, the radial $n_{r}$ and $l$ orbital quantum numbers and parameters
$D,a,R_{0}$ are also identified in detail. Finally, the bound state energies
and the corresponding normalized hyper-radial wave functions for the neutron
system of the a $^{56} Fe$ nucleus are calculated in $D=2$ and $D=3$, as well
as the energy spectrum expressions of other highest dimensions are identified
by using the energy spectrum of $D=2$ and $D=3$.
