The singular Sturm-Liouville boundary value problem
(x y^{\prime})^{\prime}=\lambda x y,\quad0\lt x\lt 1,
y\ {\text{and}}\ y^{\prime}\,\text{bounded as}\,x\to0,\quad y(1)=0
has eigenfunctions \phi_{n}(x)=J_{0}(\sqrt{\lambda_{n}}x).
(a) Show that the \phi_{n} satisfy the orthogonality relation
\int_{0}^{1}x\phi_{m}(x)\phi_{n}(x)d x=0,\quad m\neq n (21)
with respect to the weight function r(x) = x.
(b) Given a function f with f and f ′ piecewise continuous on 0 ≤ x ≤ 1, find the coefficients c_{n} such that
f(x)=\sum_{n=1}^{\infty}c_{n}J_{0}{\Big(}{\sqrt{\lambda_{n}}} x{\Big)}. (22)