Why is differential Galois theory not widely used? E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. Whether it is of closed form or not. see)
My question is why differential Galois theory is not widely used in differential geometry. It is plausible that we can solve some problems of differential/integral geometry using this set of theory.
I have read some answers provided here like Why do we need admissible isomorphisms for differential Galois theory? and other stuffs. I have read Kaplansky's and Buium's books. My question follows:
So what is the major 'pullback' in this theory that prevents its wide application to other situations rather than discrete geometry (e.g. Diophantine geometry)?
My original question on Mathematics Stack Exchange is: Why differential Galois theory is not widely used? which yields no satisfying answers.
 A: The theory of differential Galois theory is used, but in algebraic, not differential geometry, under the name of D-modules. A D-module is an object that is somewhat more complicated than a representation of the differential Galois group, in the same way that a sheaf is a more complicated than just a Galois representation, but I think it is cut from the same cloth. A D-module describes not just the solutions of a differential equation but also how they behave at singularities.
D-modules are used in many different algebraic geometry situations.
While differential Galois theory may seem analytic it is actually much more algebraic. For instance, in analysis and differential geometry you tend to care how large things are, while in algebra you don't, and differential Galois theory says nothing about size. In algebra you hope for exact solutions, while in analysis approximate solutions are usually good enough, and differential Galois theory good for describing exact solutions. In differential geometry you often have great freedom in gluing together local pieces to get a global structure, where in algebra local pieces are rigid and hard to glue together, and differential Galois theory describes rigid structures where one tiny piece controls everything.
A: As indicated by KConrad in his comments, differential Galois theory is used in the part of transcendental number theory that tries to establish algebraic/linear independence of values of special functions at algebraic numbers. Examples are given by the theorems of Siegel-Shidlovski, Nesterenko, etc.
Roughly speaking, its rôle is  to guarantee that an auxiliary function constructed in the arithmetic part of the proof is non-zero.
Contrary to what he writes, there is  a nonlinear differential Galois theory, namely Malgrange's theory of the differential groupoid of a foliation. It is not widely used in transcendental number theory  for the moment.
A reason why differential Galois theory does not explicitly appears in differential geometry is that DGT works in the framework of differential fields, and smooth functions on a connected open subset of $\mathbf R^n$ do not form a field (unless $n=0$).
However, as explained in the book of Thomas Hawkins (Emergence of the theory of Lie groups), the works of Lie were explicitly motivated by generalizing Galois theory to differential equations.
