I am considering the following boundary value problem: $$-\frac{\mathrm{d}}{\mathrm{d}x} \left[ a(x) \frac{\mathrm{d}}{\mathrm{d}x}(u(x)) \right] + c(x)u(x) = f(x),$$ where $x \in [0,1]$ and $u(0) = u(1) = 0.$
I searched through the boyce and diprima differential equations book but did not find any physical interpretation to the differential equation above with the given boundary conditions. From what I've seen, the equation above arises as a result of solving PDEs. I'm looking for a physical interpretation for the diff eq itself.
More specifically, it'd be great if someone can point me to a reference which specifies what $a(x),c(x),f(x),u(x)$ can mean. I already have an idea of how my $a(x)$ will be represented. I intend to play around with $c(x),f(x)$ to obtain interesting looking solutions $u(x)$. However, I don't want to just blindly play around with $c(x),f(x)$ not knowing what they mean.
Suggestions appreciated.