Perhaps you are looking for something deeper, but right there at the beginning of Hotta, Takeuchi, and Tanisaki's book on D-mods in the introduction is the connection to Linear PDEs.
I quote:
Therefore, systems of linear partial differential equations can be identified with the
D-modules having some finite presentations like (0.0.3), and the purpose of the theory
of linear PDEs is to study the solution space HomD(M, O). Since the space
HomD(M, O) does not depend on the concrete descriptions (0.0.2) and (0.0.3) of
M (it depends only on the D-linear isomorphism class of M), we can study these
analytical problems through left D-modules admitting finite presentations. In the
language of categories, the theory of linear PDEs is nothing but the investigation
of the contravariant functor HomD(•, O) from the category M(D) of D-modules
admitting finite presentations to the category M(C) of C-modules.