The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$ x^2 y'' + a x y' + b y = 0 $$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but tried a random sample) came up with no actual applications but just lots of vague "This is really important."s. The closest actual application was on Wikipedia's page, which says:
The second order Euler–Cauchy equation appears in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates.
Is there a more direct application of this ODE? Ideally, I'd like something along the lines of deriving the corresponding ODE with constant coefficients from considering springs or pendula.
My motivation is pure and simple that I'd like to be able to say something in class a little more motivating than: "We study this ODE simply because we can actually write down a solution, and it's quite amazing that we can do so."