Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 6 / a / ii / Solution

First move the order parameter term to the right-hand side:Here $p=x^2$, $r=x$, so the integrating factor obeysChoose the irrelevant positive constant $C=1$. Multiplying the equation by $1/x$ givesThus a Sturm-Liouville theory eigenvalue formulation with the requested eigenvalue isThis is the Sturm-Liouville formulation with Bessel order as eigenvalue. The spectral parameter is the squared order of the Bessel function, with $\beta$ treated as fixed. This differs from the also familiar formulation in which ${\beta }^2$ is the eigenvalue.

The factor $1/x$ is singular at the origin. Consequently the integral defining $w$ cannot literally use lower limit zero; take a positive reference point $x_{\ast }>0$, obtain $w=x_{\ast }/x$, and absorb $x_{\ast }$ into the normalization. The differential identity holds for $x>0$. Boundary conditions and admissibility at zero and at the outer endpoint are additional ingredients needed to make a particular self-adjoint spectral problem.