I was trying to solve the PDE $-u_{xx} + 5u_{yy} = f(x,y)$ in the domain $\Omega = (0,2) X (0,3)$ with boundary conditions $\,u|_\Omega = 0$ using fourier series.
I could use separation of variables method to get the following
$$u(x,y) = g(x)h(y)$$
$$\frac{g_{xx}}{5g} = \frac{h_{yy}}{h} = -p^2$$
and then i got
$$for \,\,\,p = \frac{n\pi}{2\sqrt 5} \,\,,g(x) = \sum_{i=0}^n B_n sin(\frac{n\pi x}{2})$$ and $$for\,\,\,p = \frac{n\pi}{3} \,\,,h(y) = \sum_{i=0}^n C_n sin(\frac{n\pi y}{3})$$
How can p be different for the $g(x)$ and $h(y)$ ?
If I have to choose an agreeable $p$ value what should it be ?