Let $k$
be a field of characteristic zero, $f(X,Y),
g(X,Y)\,{\in}\,k[X,Y]$, $g(X,Y)\,{\notin}\,(X,Y)$ and
$d\,{:=} g(X,Y)\,{\partial}/
{\partial X}+f(X,Y)\,{\partial}/{\partial Y}$.
A connection is established between the $d$-simplicity of the local ring
$k[X,Y]_{(X,Y)}$
and the transcendency of the solution in $tk[[t]]$ of the algebraic differential
equation $g(t,y(t))\,{\cdot}\,
({\partial}/{\partial t})y(t)=f(t,y(t))$.
This connection is used to obtain some interesting results in the theory
of the formal power series and to construct new examples of differentially simple rings.
