question about heat conduction. I was casually reading through an ebook and came across the following:
Could you tell me where this last equation comes from? I can follow the solution to the problem but why is it one dimensional and hence an ODE?
When would we need to use a Pde and what would be the solution to that problem?
The general heat equation is indeed a partial differential equation, as the temperature $T$ depends on both spatial and temporal coordinates: $$\alpha\nabla^2 T=\frac{\partial T}{\partial t}$$
However, we are told that we are looking at steady state, which by definition is time-independent, so $\partial T/\partial t=0$. Furthermore, the system is radially symmetric, so this means that the function $T$ can be fully specified by a single variable $r$, which is the distance from the origin. $\nabla^2$ in such an instance in spherical coordinates is then $$\nabla^2=\frac{1}{r^2}\frac{\text d}{\text dr}\left(r^2\frac{\text d}{\text dr}\right)$$
Combining these two things, the steady state solution and dependence on only one spatial variable $r$, leads to
$$\frac{\text d}{\text dr}\left(r^2\frac{\text dT}{\text dr}\right)=0$$
