Chapter 11.3, Problem 26E

Prove that the linear differential operator associated with the periodic Sturm-Liouville problem (34) is selfadjoint.

${\left(p{y}^{\prime }\right)}^{\prime }+qy+\lambda ry=0;y\left(0\right)=y\left(2T\right),y^{\prime }\left(0\right)=y^{\prime }\left(2T\right)$, $\left(34\right)$

where $p$, $p^{\prime }$, $q$ and $r$ are continuous $2T$-periodic functions with $p$ and $r$ positive.