To solve this problem, we can use the method of separation of variables. We can assume that the solution can be written as u(\phi,t) = X(\phi)T(t). Substituting this into the given equation, we get:
1/sin(phi) * d/d\phi(sin(phi) * X'(\phi)T(t)) - d^2(X(\phi)T(t))/dt^2 = -sin 2t
Rearranging and dividing by X(\phi)T(t), we get:
1/sin(phi) * d/d\phi(sin(phi) * X'(\phi))/X(\phi) = -d^2T(t)/dt^2 - sin 2t/T(t)
The left side of the equation only depends on \phi, while the right side only depends on t. Therefore, both sides must be equal to a constant, which we will call -\lambda. This gives us two separate equations:
1/sin(phi) * d/d\phi(sin(phi) * X'(\phi))/X(\phi) = -\lambda
d^2T(t)/dt^2 + (sin 2t + \lambda)T(t) = 0
Solving the first equation, we get:
X(\phi) = A_n * sin(sqrt{\lambda_n}\phi) + B_n * cos(sqrt{\lambda_n}\phi)
where \lambda_n = n^2, n = 1,2,3,...
Solving the second equation, we get:
T(t) = C_n * sin(sqrt{\lambda_n}t) + D_n * cos(sqrt{\lambda_n}t)
where C_n and D_n are constants determined by the initial conditions.
Thus, the general solution is given by:
u(\phi,t) = \sum_{n=1}^{\infty} (A_n * sin(sqrt{\lambda_n}\phi) + B_n * cos(sqrt{\lambda_n}\phi)) * (C_n * sin(sqrt{\lambda_n}t) + D_n * cos(sqrt{\lambda_n}t))
To determine if this solution exhibits resonance, we need to look at the natural frequencies, which are given by \omega_n = \sqrt{\lambda_n} = n. In this case, the natural frequencies are not equal to the forcing frequency, which is \omega = \sqrt{2}. Therefore, we can conclude that
