How to Solve this Second Order PDE with Fixed Boundaries in Y?

[phonic]

Hi All,

I try to solve second order PDE:
$\frac{\partial^2 f(x,y)}{\partial x^2}=-a^2f(x,y)$
$\frac{\partial^2 f(x,y)}{\partial y^2}=-a^2f(x,y)$
where $a >2$, $f(x,y)$ is a periodic function in x, but has fixed boundaries in y.

Is there a way to solve it? What does the solution look like? Any hints or references are welcome. thanks a lot!

 

[fluidistic]

Hi All,

I try to solve second order PDE:
$\frac{\partial^2 f(x,y)}{\partial x^2}=-a^2f(x,y)$
$\frac{\partial^2 f(x,y)}{\partial y^2}=-a^2f(x,y)$
where $a >2$, $f(x,y)$ is a periodic function in x, but has fixed boundaries in y.

Is there a way to solve it? What does the solution look like? Any hints or references are welcome. thanks a lot!

I'm also learning PDE's so take what I say with a grain of salt.
From my experience one needs to describe the region of where the PDE is evaluated/calculated. Also, telling us what are the "fixed boundaries in y" is also very important.
Edit: Your equations read $\nabla ^2 f = -2a^2 f$ where f depends on 2 spatial variables x and y.