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[en] We construct Darboux transformations for a generalized Schroedinger equation by means of the intertwining operator method. We establish a relation between first-order Darboux transformations, supersymmetry, and factorization of the Hamiltonians that are associated with our generalized Schroedinger equation. Furthermore, our methods allow for the generation of isospectral potentials, where one of the potentials has additional or less bound states than its partner. In the particular case of a conventional Schroedinger equation our generalized Darboux transformations reduce correctly to the well-known expressions.