Chapter 11.RP, Problem 10RP

Use Corollary $5$ in Section $11.8$ to estimate the distance between consecutive zeros in $\left[0,\pi \right]$ of a nontrivial solution to

${\left[\left(2+\cos x\right)y^{\prime }\right]}^{\prime }+\left(\sin x\right)y+5y=0$.

Corollary 5.

Let

$\frac{d}{dx}\left[p\frac{dy}{dx}\right]+qy+\lambda ry=0$, $a<x<b$, (15)

be a Strurm-Liouville equation, where $p$, $p^{\prime }$, $q$, and $r$ are continuous on $\left[a,b\right]$. If $\varphi$ is nontrivial solution to (15) with consecutive zeros $x_1$ and $x_2$ in $\left[a,b\right]$ and

$\lambda >\max \left\{\frac{-q_M}{r_M},\frac{-q_m}{r_m},0\right\}$,

then $x_2-x_1$, the distance between the zeroes of $\varphi$ is bounded by

$\pi \sqrt{\frac{p_m}{q_M+\lambda r_M}}\le x_2-x_1\le \pi \sqrt{\frac{p_M}{q_m+\lambda r_m}}$. (16)