  Let $O_X$ (resp. $D_X$) be the sheaf of holomorphic functions (resp. the
sheaf of linear differential operators with holomorphic coefficients) on $X$
(=the complex affine n-space). Let $Y$ be a locally weakly quasi-homogeneous
free divisor defined by a polynomial $f$. In this paper we prove that, locally,
the annihilating ideal of $1/f^k$ over $D_X$ is generated by linear
differential operators of order 1 (for $k$ big enough). For this purpose we
prove a vanishing theorem for the extension groups of a certain logarithmic
$D_X$--module with $O_X$. The logarithmic $D_X$--module is naturally associated
with $Y$. This result is related to the so called Logarithmic Comparison
Theorem.
