"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?