In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute group a second-order linear q-difference equation with rational function coefficients. This encodes possible polynomial relations among solutions of equation. our results groups several concrete equations, including colored Jones certain knot.