The Woods-Saxon potential does not give us closed-form or analytical solutions of the eigenvalue problem $$ \hat{h}_0(x_i)\psi_{\alpha}(x_i)=\varepsilon_{\alpha}\psi_{\alpha}(x_i). $$ For the harmonic oscillator in three dimensions we have closed-form expressions for the energies and analytical solutions for the eigenstates, with the latter given by either Hermite polynomials (cartesian coordinates) or Laguerre polynomials (spherical coordinates).

To solve the above equation is however rather straightforward numerically.