Problem asks me to prove Uniqueness but Im not sure how to d

Problem asks me to prove Uniqueness but I\'m not sure how to do it! From my ODE\'s and PDE\"s Class

Solution

Uniqueness arguments are identical for both both problems (in general, for this category of equations):

Let u and v are two solutions satisfying (the reduced Helmholtz) equations ) an elliptic equation and the boundary conditons (Dirichlet and Neumann).

By linearlity u-v satisfies the elliptic equation with boundary conditions zero.

Applying the maximum principle to u-v yields u-v =0 throughout. Hence the uniqueness.