# Show eigenvalues of boundary problem are real

I have the boundary problem $$(xy)''+\lambda xy=0$$ $$y(1)=0$$ $$||y'(x)||<M$$

defined on the interval [0,1], and am asked to show its eigenvalues are real.

What I have so far: I can transform the equation into a Sturm-Liouville problem: $(py^{'} )'+(\lambda r(x)-q(x) )y=0$ where $p=x^2$, $r=x^2$, $q=0$, and we learned in class that the eigenvalues of a regular Sturm-Liouville problem are real, but this isn't a regular Sturm-Liouville problem (instead we're given that the derivative is bounded). So I'm stuck.

Any suggestions?

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