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We have the laplac problem in $(0,1)^2$ with the boundary conditions $$u(x,0)=f(x) , \ u(x,1)=0, \ \ x\in [0,1] \\ u(0,y)=u_x(1,y)=0, \ \ y\in [0,1]$$ Using the separtaion of variables we get the following two problems $$X''(x)=\lambda X(x)=0, X(0)=X'(1)=0 \\ Y''(y)-\lambda Y(y)=0, \ \ Y(1)=0$$ F...