  Let $A$ be an associative algebra over a field $F$ of characteristic zero and
let $L$ be a Lie algebra over $F$. If $L$ acts on $A$ by derivations, then such
an action determines an action of its universal enveloping algebra $U(L)$ and
in this case we refer to $A$ as algebra with derivations or $L$-algebra.
  Here we give a characterization of the ideal of differential identities of
finite dimensional $L$-algebras $A$ in case the corresponding sequence of
differential codimensions $c_n^L (A)$, $n\geq 1$, is polynomially bounded. As a
consequence, we also characterize $L$-algebras with multiplicities of the
differential cocharacter bounded by a constant.
