TITLE: Why isn't differential Galois theory widely used?
QUESTION [20 upvotes]: Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation.
Why isn't differential Galois theory widely used in differential geometry? It is plausible that we can solve some problems of differential/integral geometry using this theory.
So, what is the major pullback in this theory that prevents its wide application to other fields rather than discrete geometry (e.g., Diophantine geometry)?

REPLY [4 votes]: Among others, there is a nice concrete application differential Galois theory to the  Non-Integrability of Hamiltonian Systems :
http://www.springer.com/us/book/9783034807203