Solution
= Solution
The <product rule> makes the drift term cancel:
$$
u'+\beta V'u=e^{-\beta V}y'.
$$
Consequently the transformed <eigenvalue problem> is
$$
-\bigl(De^{-\beta V}y'\bigr)'=\lambda e^{-\beta V}y,
\qquad
\boxed{p=De^{-\beta V},\quad q=0,\quad w=e^{-\beta V}.}
$$
Both $p$ and the weight $w$ are positive, as required for this <Sturm-Liouville problem>.