I am studying the method of separation of variables for the heat equation. I am struggling to understand why the separation constant takes on a positive or negative value. For instance if we apply the method of separation of variables to this equation. $$\frac{\partial u}{\partial t}= \frac{k}{r} \frac{\partial}{\partial r} (r \frac{\partial u}{\partial r} ) $$
looking for product solutions $u(r,t)=\phi(r)h(t)$ then dividing by $\phi h $ yields... $$\frac{1}{kh}\frac{dh}{dt}=\frac{1}{r \phi} \frac{d}{dr} (r \frac{d \phi}{dr})=-\lambda$$
But the following problem ends up with a positive separation constant. $$\frac{\partial u}{\partial t}= k \frac{\partial^4}{\partial x^4} $$
looking for product solutions $u(x,t)=\phi(x)h(t)$ then dividing by $k \phi h $ yields... $$\frac{1}{kh}\frac{dh}{dt}=\frac{1}{\phi} \frac{\partial^4}{\partial x^4}=\lambda$$
Why does the first problem end up with a negative constant and the second gets a positive one?
Edit: Does it have something to do with this question Why can we assume the separation constant??