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I am interested in solving the following biharmonic eigenvalue problem.

$$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - \frac{\pi}{2} \le x \le \frac{\pi}{2} & -\frac{\pi}{2} \le y \le \frac{\pi}{2} \\ & x = \phantom{-}\frac{\pi}{2} & \Psi = 0 & \dfrac{\partial \Psi }{\partial x} = 0 \\ & x = - \frac{\pi}{2} & \Psi = 0 & \dfrac{\partial \Psi }{\partial x} = 0 \\ & y = \phantom{-}\frac{\pi}{2} & \Psi = 0 & \dfrac{\partial \Psi }{ \partial y} = 0 \\ & y = - \frac{\pi}{2} & \Psi = 0 & \dfrac{\partial \Psi }{ \partial y} = 0 \end{array} $$

where $\Delta^2$ is the biharmonic operator.

$$ \Delta ^2 \Psi = \frac{\partial ^4 \Psi }{\partial x^4} + 2 \frac{\partial^4 \Psi }{\partial x^2 \partial y^2} + \frac{\partial ^4 \Psi }{\partial y^4}$$

I looked into DEigenvalues but I couldn't understand how the BCs are determined.

Can anyone help me to tell Mathematica to find the first five eigenvalues of this problem for me?

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