Ok, so you have already know how to derive the formula, the big problem now is how to solve the second order differential equation. Actually, you can find some books and read yourself which is really benefical. But here I will tell you the answer. The hoop will undergo a uniform circular motion on the radical. But on the normal, the answer should be y= normal part C1*e^w+C2*e^(-w)+ particular part (g/1+w^2)*cos(x) where c1 and c2 are arbitrary constant because you want to find the particular answer but not the normal answer. the third speaker is right if we do not consider the particular answer. Hope helps Cheers