Solution
= Solution
Divide the eigenvalue equation by $x$ and collect the first two derivative terms into a single derivative:
$$
-(xy')'-\frac yx=\lambda\frac yx.
$$
Thus one choice of <Sturm-Liouville form> is
$$
\boxed{p(x)=x,\qquad q(x)=-\frac1x,\qquad w(x)=\frac1x,\qquad 1\leq x\leq e.}
$$
Indeed $w^{-1}[-(py')'+qy]=-x^2y''-xy'-y$. Both $p$ and $w$ are positive and regular throughout this interval. The zero endpoint values make the boundary form vanish.