  In this article, the Darboux transformation for the non-isospectral AKNS
hierarchy is constructed. We show that the Darboux transformation for the
non-isospectral AKNS hierarchy is not an auto-B\"acklund transformation,
because the integral constants of the hierarchy will be changed after the
transformation. The transform rule of the integral constants will be also
derived. By this means, the soliton solutions of the nonlinear equations
derived by the non-isospectral AKNS hierarchy can be found.
