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I'm struggling with this question: we have two differentiable functions $p, q$ in some interval $I$ and the equation $(p(x)y')'+q(x)y=0$, with given solutions $u$ and $1/u$. We're asked to show $u$ solves a first-order differential equation.

What I tried: I tried two things: plugging the solutions to the equation and differentiating, and plugging the solutions and integrating (to get rid of the 2nd order derivative). It didn't get me very far, though ...

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    Yup, I made a calculation error which made me think it was the wrong approach. Oh well!2012-10-30

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We know $ \left(p u'\right)' + qu = p u'' + p' u' + qu = 0 $ and $ \left[p \left(\frac{1}{u}\right)'\right]' + \frac{q}{u} = \frac{2 p u'^2}{u^3} + \frac{p u''}{u^2} - \frac{p' u'}{u^2}+\frac{q}{u} = 0. $ Multiply the second equation by $u^2$: $ \frac{2 p u'^2}{u} + p u'' - p' u' + q u = 0. $ Subtract this equation from the first equation: $ 2 p' u' - \frac{2 p u'^2}{u} = 0. $

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    Ugh. This is exactly what I did, but I made a calculation error which made me think it wouldn't work.2012-10-30