Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 1 / a / Solution

Seek an integrating factor $\mu$ such that the differential expression becomes $-(p{y}^{\prime }{)}^{\prime }$. Its leading coefficient requires $p=\mu (1-x^2)$, while the first-derivative coefficient requires $-p^{\prime }=3x\mu$. ThusChoose the positive solution $\mu =\sqrt{1-x^2}$. The Sturm-Liouville form is thereforewithThe weight is positive on the open interval. The endpoints are singular because $p$ vanishes there, but the prescribed boundedness of $y_n,y_n^{\prime }$ makes the integration-by-parts boundary terms vanish.