1. ## partial differential equation

How to solve the attached coupled equations for $\psi$ and $u$. Please help me. Its very complex.

2. ## Re: partial differential equation

You can try separation of variables.

The given equation is $\dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}=k^2\psi$.

Let $\psi(x,y)=a(x)b(y)$ then $\dfrac{\mathrm{d}^2 a}{\mathrm{d} x^2}b+a\dfrac{\mathrm{d}^2 b}{\mathrm{d} y^2}=k^2ab$

$\dfrac{1}{a}\dfrac{\mathrm{d}^2 a}{\mathrm{d}x^2}+\dfrac{1}{b}\dfrac{\mathrm{d}^2 b}{\mathrm{d}y^2}=k^2$

This means $\dfrac{1}{a}\dfrac{\mathrm{d}^2 a}{\mathrm{d}x^2}=k^2-\dfrac{1}{b}\dfrac{\mathrm{d}^2 b}{\mathrm{d}y^2}=C$ where $C$ is a constant.

I leave the rest to you.