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.

lineardifferential equations inonevariable. Specialists in differential equations tend to care about more than that (e.g., PDE, nonlinear equations), and the aspect of solvability addressed by differential Galois theory is largely of no interest to them even for linear differential equations. Even ordinary Galois theory is not really of much importance to analysts needing roots of a polynomial in C. Nobody really cares about being able write roots in a very restrictive form related to solvable Galois groups. It is mostly of historic interest. $\endgroup$ – KConrad Apr 3 '15 at 2:546more comments