Over $\mathbb{A}^n$, it is easy to construct D-modules by writing down an explicit linear system of PDE's and then writing a presentation of the associated D-module
$$
\mathcal{D}^n \xrightarrow{} \mathcal{D}^m \to \mathcal{M} \to 0
$$
Is there an analogous process for constructing D-modules over projective space as an explicit system of differential equations?



How can I construct D-modules over projective space using explicit differential equations?
You can construct $\mathcal{D}$ modules on projective space (or actually any space) by precisely the same process: write down a system of PDE's (using differential operators on the whole of $\mathbb{P}^n$) and consider the corresponding quotient. The key caveat is that on a general space, not every $\mathcal{D}$-module will be of this form: some of them can be defined using systems on different coordinate patches which are consistent, but not globally (I think a degree 0 line bundle on an elliptic curve should give a counterexample). However, $\mathbb{P}^n$ is special, and every $\mathcal{D}$-module can be defined this way (by the Beilinson-Bernstein theorem). 


