The book "D-Modules, Perverse Sheaves, and Representation Theory " by Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki is the perfect source for this topic. The introduction gives a very nice (and elementary) explanation how representation theory of D-modules and symstems of partial differential equations are related. I just give a very nice excerpt from the introduciton of the book.

Let $X$ be an open subset of $\mathbb{C^n}$ and $\mathcal{O}$ the commutative ring of complex analytic functions defined on $X$.
Let $D$ be the set of partial differential operators with coefficients in $\mathcal{O}$, whose elements are thus of the form $\sum\limits_{i_1,...,i_n}^{\infty}{f_{i_1,...,i_n} (\frac{\delta}{\delta x_1})^{i_1} ... (\frac{\delta}{\delta x_n})^{i_n}} $.

Let $P \in D$ and consider the partial differential equation $Pu=0$ and $M$ the D-module $M=D/DP$.
We then have $Hom_D(M,\mathcal{O}) \cong \{f \in \mathcal{O} | Pf=0 \}$.
This shows that the set of analytic solution of $Pu=0$ is isomorphic to a $Hom$-space, which are the natural objects of study of representation theory (representation theory can be summarized more or less as the study of representations of rings and their Hom-spaces).