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Assume that $p(x)<0$ for all real numbers and y(x) is a solution of the DE $y'+p(x)y=0$ that is not identically zero. I need to prove that y can cross the x axis at most once.

I don't understand the answer below, any other help, please?

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Rough idea: between two consecutive zeroes of $y$, $y'$ must vanish. But $y'=-p(x)y(x)$, which has constant sign between consecutives zeroes of $y$.

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    By Rolle's theorem, $f(a)=f(b)=0$ implies the existence of$a$critical point between $a$ and $b$. Then you can derive a contradiction with the DE.2012-12-12