I think the short answer is that the boundary conditions are very different in the two cases. In case (i) you are given the value of u(x,0) for all x, but in case (ii) you only have the value at one point, u(1,0). In case (i), the boundary condition is strong enough to determine the solution uniquely. But in case (ii) there is not enough information for that, and the general solution has to include an arbitrary function of x^2+y^2.