My text says the following
$$ \frac{\mathrm d}{\mathrm dx}\left(x^2 \frac{\mathrm dy}{\mathrm dx}\right) + \lambda y = 0,\;\;\;0\le x\le 1,\; y(1)=c\ge0$$
is an "eigenvalue problem".
I don't understand the significance of calling it an eigenvalue problem. This equation can be solved by expanding the leftmost expression and recognising that it has the form of Euler's equation. I'm really struggling to see what's "eigenvalue" about it compared to other second order homogeneous linear differential equations I've seen in the past.
This is in the context of Sturm-Louiville systems with the more general equation $$ \frac{\mathrm d}{\mathrm dx}\left(p(x) \frac{\mathrm dy}{\mathrm dx}\right) + (q(x)+\lambda w(x) )\,y = 0 $$