How to solve the attached coupled equations for $\displaystyle \psi$ and $\displaystyle u$. Please help me. Its very complex.
You can try separation of variables.
The given equation is $\displaystyle \dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}=k^2\psi$.
Let $\displaystyle \psi(x,y)=a(x)b(y)$ then $\displaystyle \dfrac{\mathrm{d}^2 a}{\mathrm{d} x^2}b+a\dfrac{\mathrm{d}^2 b}{\mathrm{d} y^2}=k^2ab$
$\displaystyle \dfrac{1}{a}\dfrac{\mathrm{d}^2 a}{\mathrm{d}x^2}+\dfrac{1}{b}\dfrac{\mathrm{d}^2 b}{\mathrm{d}y^2}=k^2$
This means $\displaystyle \dfrac{1}{a}\dfrac{\mathrm{d}^2 a}{\mathrm{d}x^2}=k^2-\dfrac{1}{b}\dfrac{\mathrm{d}^2 b}{\mathrm{d}y^2}=C$ where $\displaystyle C$ is a constant.
I leave the rest to you.