I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more clarity than typing it here. It is all about solution of a quaternion differential equation
Here the main issue for me is, I couldnt find a solution for q(t) in time varying angular velocity. What we have here as solution is for a constant angular velocity as in (31) and (32). What could be the solution if the angular velocity is not a constant(it varies with time)?. I have a feeling that equation (39) in paper here is a solution of this issue. But it is not a solid proof. It would be nice if some one suggests the proof of it or even a proper reference link for it. Material is not available that is the main issue I am facing