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