I am working my way through a textbook (Richard Haberman fourth edition) on the heat equation as an example of applied partial differential equations. I am not familiar with the concept of a separation constant an it keeps coming up in the derivations. Forgive me I am a neuroscience major not a math major.
For instance I am on chapter two, we are discussing Laplace's Equation for heat flow in a rectangular surface. We are given this equation $$\frac{1}{h}\frac{\partial^2 h}{\partial x^2}=-\frac{1}{\phi}\frac{\partial^2 u}{\partial y^2}=\lambda, $$
where \lambda is the eigenvalue or separation constant of this gradient. I understand an eigenvalue in the context of linear algebra (which I understand well enough) and I am willing to accept that functions are infinitely indexed vectors but I am still confused as to how I can just pull that separation constant out of the air. What conditions need to be met to make this assumption?
Edit: Here is the page in my text this is taken from, maybe there is relevant information I am not including.
The point is that
$$f(x) = \frac{1}{h }\frac{\partial^2 h}{\partial x^2}$$
is independent of $y$, while
$$g(y)=-\frac{1}{\phi}\frac{\partial^2 u}{\partial y^2} $$
is independent of $x$. So you are in a situation where
$$ f(x) = g(y), \ \ \ \text{for all }x, y.$$
This implies that $f, g$ are both constant functions. For example, choose $y=0$, then $f(x) = g(0)$ for all $x$. So $f(x)$ is a constant function. Similar for $g$.
Thus $f(x) = g(y) = \lambda$.