So, my problem is as follows:
$ U_{xx} +U_{yy}=0; U(x,0)=0, U_x (0,y)=0 ,U_x (L,y)=0 $
I need to use Separation of Variables, which I did, and got:
$ U(x,y) = (Ae^{\sqrt{\lambda}x}+Be^{-\sqrt{\lambda}x})(C\cos{\sqrt{\lambda}y}+D\sin{\sqrt{\lambda}y}) $
for ${\lambda}>0$. My question is how to plug in the boundary conditions, and why this first solution does/doesn't satisfy the conditions.