  In this paper we utilize the covariance of Ricatti equation with respect to
linear fractional transformations to define classes of conformally equivalent
second order differential equations. This motivates then the introduction of
fractional Darboux transformations which can be recognized also as generalized
Cole-Hopf transformations. We apply these transformations to find Schrodinger
equations with isospectral potentials and to the linearization of some new
classes of nonlinear partial differential equations.
