Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 14A / b / i / Solution

Multiply the equation by $x^2$ and combine the derivative terms:Thus the self-adjoint form isThe 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.