All in all, it is surprising and kind of remarkable how much knowledge about differential equations you can derive by doing algebra. I hope someone liked the thread and if requested i can post my Bachelor thesis for a very accessible read. Thanks for anyone who read this far.

Now the calculation of the differential Galois group is an ongoing field of research and is in general really really hard( altough there exists a complete algorithm for it), but for special kinds of equations namely the Fuchsian differential, the Galois group is really just the closure of its Monodromy group under the Zariski topology, which gives us a way to understand the solutions of these equations by looking at the poles of its solutions.

And if you are like me and begin to wonder, ok fine and dandy, but what about partial differential operators? Is that theory so much harder in comparison to the ordinary case? And the answer is that unlike the analysis of PDE's the partial case isnt at all harder, and in fact all reduceable back to the ordinary case.

More than that even. We can extend that result to basicly say, every restriction to the expressibility of the solutions one to one translates to solvability restrictions of the Galois group. Kolchin proved a myriad of theorems, for example that the solvability by integrals translates to the special triangularity of the Galois group and many many more.