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.