I'm following the method in a paper to solve the following partial differential for $\psi(x,y)$ where $G_x$ and $G_y$ are constants,
$$ G_x(\frac{\partial^2\psi}{\partial x^2})+G_y(\frac{\partial^2\psi}{\partial y^2})=0 $$
for this rectangle. What kind of equation is this? I know it's a Laplace equation but is there a more specific category? Is it a homogeneous equation? The boundary conditions for it are
$$ \frac{\partial\psi}{\partial x}=2y $$
at $x=0,a$ and
$$ \frac{\partial\psi}{\partial y}=0 $$
at $y=\pm b$. To find $\psi(x,y)$, the authors assume that
$$ \psi(x,y)=\sum_{n=1,3,5...}^{\infty}{f_n(x)\sin(\xi_n y)} $$
where $f_n(x)$ is a series of undetermined functions and $\xi_n=\frac{n\pi}{2b}$. What kind of equation is the one that they assumed for $\psi(x,y)$? A Fourier series of some sort? They state in the paper that they avoid using hyperbolic functions but in another potential equation for $\psi(x,y)$ that I found in another paper that deals with the same problem is,
$$ \psi(x,y)=xy-\sum_{m=0,1,2...}^{\infty}{c_n\sin(\kappa_nx)\sinh(\kappa_ny)} $$
where $\kappa_n=\frac{(2n+1)\pi}{2a}$. This is also a Fourier series, am I right? Is there a reason why one would want to use one equation for $\psi(x,y)$ over the other?
This is Laplace's equation only if $G_x=G_y\not =0 $. If $G_x\cdot G_y>0$ it is an elliptic equation and if $G_x\cdot G_y<0$ it is hyperbolic equation. In any case it is homogeneous.
The solution suggested is a Fourier series used in the method "separation of variables" for this kind of problems. The type of Fourier series used depends on the problem (equation and boundary conditions), so for different problems one should consider different types of Fourier series.