# Solve a Differential Equation

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.

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 $y$.
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. –  Siminore Dec 12 '12 at 17:34