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In this paper, the approximate analitical solutions of the hyper-radialSchr\"{o}dinger equation are obtained for the generalized Wood-Saxon potentialby implementing the Pekeris approximation to surmount the centrifugal term. Theenergy eigenvalues and corresponding hyper-radial wave functions are found forany angular momentum case via the Nikiforov-Uvarov (NU) and Supersymmetricquantum mechanics (SUSY QM) methods. Hence, the same expressions are obtainedfor the energy eigenvalues, and the expression of hyper-radial wave functionstransformed each other is shown owing to these methods. Furthermore, a finitenumber 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 energiesand the corresponding normalized hyper-radial wave functions for the neutronsystem of the a $^{56} Fe$ nucleus are calculated in $D=2$ and $D=3$, as wellas the energy spectrum expressions of other highest dimensions are identifiedby using the energy spectrum of $D=2$ and $D=3$.

Bound states of the $D$-dimensional Schr\"{o}dinger equation for the generalized Woods-Saxon potential