A generalized method for the Darboux transformation

A method is presented to obtain the change in potential and relevant wavefunction of a linear system ordinary differential equations containing spectral parameter, when that perturbed finite number discrete eigenvalues are added or removed from spectrum. Some explicit formulas derived for those changes by introducing certain fundamental integral corresponding unperturbed systems. This generalized applicable unified manner on wide class contrast standard Darboux transformation, which specific particular it applies. comparison provided some special cases between this transformation. In particular, bound state spectrum, transformation full-line Schr\"odinger equation, where resemble half-line equation. The theory illustrated with examples.