Consider the problem:
$$ \Delta^2 u = f$$
on the square domain $U=(0,1)\times(0,1)$ with boundary conditions:
$$ u(x,y)=\Delta u(x,y) = 0$$
for $(x,y) \in \partial U.$
I try to solve it with the hint that i should use separation of variables but i can not do it. I write $u(x,y) = X(x) \cdot Y(y)$ and i plug this in the equation $ \Delta^2 u = \lambda u$ to get $$ X^{(4)} +2X^{(2)}Y^{(2)}+Y^{(4)} = \lambda XY$$ and here i am stuck. Any help really appreciated.