I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When I was doing a perusal on "A primer of algebraic D-modules by S. C. Coutinho" the justification on the importance of D-modules; they provide an algebraic tool towards the solution of differential equations. This is the story I always hear! Do someone have a reference or more information about D-modules and algebraically solution of PDEs ?. 



D-modules and Algebraic Solutions of PDEs 
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.
> 
> 
> 


