Auto-Balanced Rotor MatLab

I am looking for someone that can implement a system of coupled second order differential equations in MatLab. I have tried myself and almost got to the solution, however i can not come to the right solution. For reference I have tried to follow along with a paper written on the matter and as I mentioned, I do not get the right outcome. I suppose my problem is in the way I solve the coupled equations.
If someone is out there that can handle this easily, you're welcome to reply. I will provide the paper and expect the solutions to match.