Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 17B / b / Solution

With $q=p^{\prime }$ and $r=0$, the equation isMultiply by the real eigenfunction $y$ and integrate over $[a,b]$. The clamped boundary conditions $y=y^{\prime }=0$ at both endpoints eliminate all boundary terms, soSince $p(x)<0$, the right side is a sum of nonnegative terms. It cannot vanish for a nonzero eigenfunction: if $y^{\prime \prime }=y^{\prime }=0$, the boundary conditions force $y=0$. Thus