Solution
= Solution
Multiply the equation by $x^2$ and combine the derivative terms:
$$
(x^3y')'+(1+\lambda)xy=0.
$$
Thus \b[the self-adjoint form] is
$$
\boxed{-(x^3y')'-xy=\lambda xy,\qquad p=x^3,\quad q=-x,\quad w=x.}
$$
The <Dirichlet boundary conditions> remain at $1$ and $e$, and the weight $w=x$ is positive on this interval, as required for the <Sturm-Liouville problem>.