Zeros of Euler's equation, y''+(k/x^2)y=0

"Show that every nontrivial solution of has an infinite number of positive zeros if and only finitely many positive zeros if ."

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I set (for some constant M), differentiated twice and put it back into the equation, which gives . So, and solves .

The Wronskian seems to be identially nonzero, so then every solution of can be written as .

The "finitely many positive zeros if " part follows, but I'm not sure about the "infinite number of positive zeros of " part. Obviously the exponents are complex numbers that avoid the real and imaginary axes in that case.

Any other approaches?

Re: Zeros of Euler's equation (y''+(k/x^2)y=0)

I solved it, I think. could be written as the composition of sines and cosines of (the monotonic function) when with a preceding (monotonic) factor , so should have infinitely many solutions in that case.

It turns out that makes the Wronskian vanish, so with a bit of effort, and can be found, which can be shown to be linearly independent solutions in this case. The general solution when , then, has finitely many zeros in .

I think this was somehow a stupid and overly lengthy approach to the problem, though, so suggestions for alternatives are welcome!