I've been studying asymptotic behavior on Ordinary Differential Equations. While doing some excercises I found out one excercise which has had me thinking for a while, so I am asking humbly for your help. The statement is the following:
Consider the second order differential equation \begin{equation} y''+p(x)y=0 \end{equation} and its perturbed equation \begin{equation} z''+(p(x)+q(x))z=0 \end{equation} where $p(x)$ and $q(x)$ are continous functions on the interval $[x_0,\infty)$. Show that if all the solutions of the first equation given are bounded and $\int^\infty |q(x)|dx<\infty$, then all the solutions of the second equation given are bounded in $[x_0,\infty)$.
I really would appreciate some help :).