- #1

Combinatus

- 42

- 1

## Homework Statement

Show that every nontrivial solution of [itex]y''+\frac{k}{x^2}y=0[/itex] (with [itex]k[/itex] being a constant) has an infinite number of positive zeros if [itex]k>1/4[/itex] and only finitely many positive zeros if [itex]k\le 1/4[/itex].

## Homework Equations

## The Attempt at a Solution

I set [itex]y=x^M = e^{M \log{x}}[/itex] (for some constant M), differentiated twice and put it back into the equation, which gives [itex]M=\frac{1\pm \sqrt{1-4k}}{2}[/itex]. So, [itex]y_1 = x^{\frac{1}{2} (1+\sqrt{1-4k})}[/itex] and [itex]y_2 = x^{\frac{1}{2} (1-\sqrt{1-4k})}[/itex] solves [itex]y''+\frac{k}{x^2}y=0[/itex].

The Wronskian seems to be identially nonzero, so then every solution of [itex]y''+\frac{k}{x^2}y=0[/itex] can be written as [itex]y = C_1 x^{\frac{1}{2} (1+\sqrt{1-4k})} + C_2 x^{\frac{1}{2} (1-\sqrt{1-4k})}[/itex].

The "finitely many positive zeros if [itex]k\le 1/4[/itex]" part follows, but I'm not sure about the "infinite number of positive zeros of [itex]k>1/4[/itex]" part. Obviously the exponents are complex numbers that avoid the real and imaginary axes in that case.

Any other approaches?