Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 1 / ii / a / Solution

Divide the eigenvalue equation by $x$ and collect the first two derivative terms into a single derivative:Thus one choice of Sturm-Liouville form isIndeed $w^{-1}[-(p{y}^{\prime }{)}^{\prime }+qy]=-x^2{y}^{\prime \prime }-x{y}^{\prime }-y$. Both $p$ and $w$ are positive and regular throughout this interval. The zero endpoint values make the boundary form vanish.