Darboux transformation

The aim of this study is to generalize and apply the Darboux transformation to non-analytical solvable systems. The non-linear systems which cannot be solved analytically, may have been reduce to solvable cases by using the Darboux transformation. Generally, the equations which in the form of Riccati equations or Sturm-Liouville equations. The eigenvalue and eigenfunctions of the various systems have been found through solving corresponding equations. Analytically solvable potentials' spectrums have been obtained by applying the Darboux transformation to the supersymetric approach. Finally, the new solutions have been compared with known results. Key Words : Darboux Transformation, Riccati Equation, Sturm - Liouville Equation, Schrödinger Equation, Supersymetric WKB Method