For u(x,y) defined on the square domain of 0

For u(x,y) defined on the square domain of O S x S I and O y S l, consider the PDE with the boundary conditions (u and u denote au/x and u/y, respectively) (i) = O (ii) O (iii) u (x, O) = 2 (iv) u (x, l) = 2 + cos(3Jt x) (a) Test the solvability condition on the system. Based on it, which of the following is true? (I) The system has no solution. (II) The system has a unique solution. (III) The system has multiple solutions. If your answer is (I), no need to proceed further. Otherwise, proceed to Part (b). (b) Find the solution(s) of the system. If your answer for Part (a) is (III), please clearlywrite out what those multiple solutions are. For this part of the problem, we expect a closed-form analytic solution that consists of only a finite number of terms and without any unevaluated integrals. Expect a deduction if the requirement is not met.