  The optimization of the usual entropy $S_1[p]=-\int du p(u) ln p(u)$ under
appropriate constraints is closely related to the Gaussian form of the exact
time-dependent solution of the Fokker-Planck equation describing an important
class of normal diffusions. We show here that the optimization of the
generalized entropic form $S_q[p]=\{1- \int du [p(u)]^q\}/(q-1)$ (with
$q=1+\mu-\nu \in {\bf \cal{R}}$) is closely related to the calculation of the
exact time-dependent solutions of a generalized, nonlinear, Fokker Planck
equation, namely $\frac{\partial}{\partial t}p^\mu= -\frac{\partial}{\partial
x}[F(x)p^\mu]+D \frac{\partial^2} {\partial x^2}p^\nu$, associated with
anomalous diffusion in the presence of the external force $F(x)=k_1-k_2x$.
Consequently, paradigmatic types of normal ($q=1$) and anomalous ($q \neq 1$)
diffusions occurring in a great variety of physical situations become unified
in a single picture.
