Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 1 / b / i

Expanding the general Sturm-Liouville form givesMatching the second-derivative coefficients requires $p/w=x^{-4}$. Matching the first-derivative coefficients then gives $p^{\prime }/p=-2/x$. Up to a common positive normalization of $p,q,w$, one may therefore takeIndeed $-x^{-2}(x^{-2}{y}^{\prime }{)}^{\prime }-ay=-x^{-4}{y}^{\prime \prime }+2x^{-5}{y}^{\prime }-ay$. The left endpoint is singular in these coordinates. Its self-adjoint interpretation is supplied by the regular transformed problem in part (ii), rather than by assuming that $p(0)$ is finite.