  In this paper, we establish the Composition-Diamond lemma for
$\lambda$-differential associative algebras over a field $K $ with multiple
operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free
$\lambda$-differential Rota-Baxter algebras. In particular, linear bases of
free $\lambda$-differential Rota-Baxter algebras are obtained and consequently,
the free $\lambda$-differential Rota-Baxter algebras are constructed by words.
