Abstract: A generalized Drinfel'd-Sokolov-Wilson (DSW) equation and its Lax pair are proposed. A Darboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral
problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux
transformations, we give some explicit solutions of the generalized DSW equation and DSW equation such as rational solutions, soliton solutions, periodic solutions.

Key words: Drinfel'd-Sokolov-Wilson equation, Darboux transformation, explicit solutions