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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."

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I've only ever come across it when solving Laplace's equation in cylindrical coordinates by separation of variables. – j.c. Jan 22 '10 at 13:28
Frankly, solving Laplace's equation is already rather significant application. I agree with you that it'd be nice to have more, but don't sweep that one under the rug! – Tom LaGatta Jan 22 '10 at 15:06
I don't want to sweep Laplace's equation under the rug - far from it - but it's too long from that to this ODE for this particular course. If you think about the derivation of the ODE with constant coefficients from considering the mechanics of a spring and compare that with deriving the Euler-Cauchy from Laplace's equation (a PDE!) which you first need to motivate from physics ... it's too long! Hence I'm looking for something quicker. – Loop Space Jan 22 '10 at 15:26

Well, it is the simplest case of a second order equation with regular singular points, so it is a pretty good example in trying to figure out what to expect from the general case.

In fact, in a way, it contains essentially all the complexities of the general case (as long as you care about what happens locally at the singular point), which is something quite nice for a 'simplest example'!

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Well, if your purpose is simply to have a direct mechanical example motivating the introduction of this equation in the class, you can use the equation describing time-harmonic vibrations of a thin elastic rod: $$ \frac{\partial}{\partial x}\left(E(x)\frac{\partial u}{\partial x}\right)+\rho\omega^2u=0 $$ in which $E(x)$ is Young's modulus, $\rho$ is the material density and $\omega$ the angular frequency of vibrations (see e.g. Graff's "Wave motion in elastic solids"). Now, if you assume that the rod is made of inhomogeneous material such that $E(x)=E_0x^2$ (a bit unusual, but very possible) then you'll get precisely your equation with $a=2$ and $b=\rho\omega^2/E_0$.

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While probably not the application you were looking for, they are of pedagogical/theoretical significance for being (explicitly solvable) differential equations for which the supposition of a non-zero real analytic solution (i.e., solving by Taylor series) fails quite dramatically. This seems to fulfill the "say something interesting about them" clause of the question (though I should note that I don't give them much emphasis in my DEs class).

And, as Mariano points out, they're a gateway set of D.E.s to understanding what's going on in much more complicated scenarios -- the vanishing set of the leading term of a linear PDE (e.g., our Euler equation's regular singular point) defined on a manifold cuts out the characteristic variety in the manifold's cotangent bundle. As I understand it (and I'm talking a little over my head here), generalizing the solvability of Cauchy-Euler equations to this much more general setting via the characteristic variety is/was a prominent goal of Algebraic Analysis and the theory of D-modules.

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A very late answer.

I think I have a nice application which came out of a different question. I was looking for a nice 2nd order example coming from finance for my ODE class. The one I got is the equation for the Perpetual American Option (Black-Scholes time independent) which is an Euler equation.

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Can you add more details? – Amir Sagiv May 12 at 19:49

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