Chapter 11.RP, Problem 9RP

Let $\varphi \left(x\right)$ be a nontrivial solution to

${\left[\left(1-e^{-\text{x}}\right)y^{\prime }\right]}^{\prime }+\left(\frac{x+1}{x}\right)y=0$

on $\left(0,\infty \right)$. Use the Picone comparison theorem to estimate the distance between consecutive zeros of $\varphi \left(x\right)$ when $x$ is large.