Why derive Regular Singular Points?

[freydawg56]

I am currently studying a great text

Elementary Differential Equations and Boundary Valued Problems 9th edition;
and we have come to chapter 5 and are studying Ordinary Points, Singular Points, and Irregular Points. (get the point?)

Anyway, I did see these mentioned,,

this Bessel equation:

x^2 y`` + x y` + (x^2 - v^2) = 0

and the Legendre equation:
(1 - x^2) y`` -2x y` $$\alpha$$ ($$\alpha$$ + 1) = 0

and since they have their own names they do "seem" important.

This is just one chapter of my studies this semester but would anyone care to inform me of any physical relevance of these equations to the real world? I'm sure they help in some way, and I suppose I could google it, but this is the physics forums and I'm sure you guys love this stuff as much as I enjoy typing this out right now. so PF's help me please!

 

[HallsofIvy]

The Bessel and Legendre equations are what you get when you write $\nabla^2 f(x, y, z)= 0$ in cylindrical and spherical coordinates, respectively.

And that is important because the "Laplacian", $\nabla^2 f$ is the simplest second order differential operator in 3 dimensions that is "invariant under rigid motions"- that is, rotating or translating the coordinates does not change its form. And because of that, it shows up in many physical problems. For example:
potential energy: $\nabla^2 f(x,y,z)= 0$
the wave equation: $\nabla^2 f(x,y,z,t)= \frac{\partial^2 f}{\partial t^2}$
the heat or diffusion equation: $\nabla^2 f(x,y,z,t)= \frac{\partial f}{\partial t}$