I tried the tip I received from the teacher but I can't solve this exercise, even that it looks simple.
As $y(x)$ is the non-zero solution to the Airy equation
\begin{equation} y''+q(x)y=0,\end{equation}
where $q(x)>0,\forall x$ and
\begin{equation} \int\limits_0^\infty q(x)dx=\infty, \end{equation}
show that $y(x)$ has an infinite amount of positive roots.
Tip I received: Supose, by contradiction, there is a $x_0$ such that $y(x_0)=0$ and $y(x)\neq 0$ for $x>x_0$. Show that there is $x_1>0$ such that $y'(x_0)$ and $y'(x_1)$ has opposite signs so there's a root larger than $x_0$. To prove the existence of $x_1$ one can integrate by parts the Airy equation $y''/y=-q(x)$.