Question 11.4.2: The singular Sturm-Liouville boundary value problem −(xy′)′ ......
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)
(a) From the differential equation, we see that p(x) = x, q(x) = 0, and r(x) = 0. Thus the orthogonality of the eigenfunctions with weight r(x) = x is a direct consequence of Theorem 11.2.2.
(b) While Theorem 11.2.4 applies only when the eigenfunctions are normalized, multiplying equation (22) by x J_{0}{\Big(}{\sqrt{\lambda_{m}}}x{\Big)} and integrating term by term from x = 0 to x = 1, we obtain
\int_{0}^{1}{x f(x)}J_{0}\biggl(\sqrt{\lambda_{m}} x\biggr)\,d x=\sum_{n=1}^{\infty}c_{n}\int_{0}^{1}{x J_{0}\biggl(\sqrt{\lambda_{m}} x\biggr)}\,J_{0}\biggl(\sqrt{\lambda_{n}} x\biggr)\,d x. (23)
Because of the orthogonality condition (21), the right-hand side of equation (23) collapses to a single term; hence
c_{m}=\frac{\displaystyle\int_{0}^{1}x f(x)\,J_{0}\Bigl(\sqrt{\lambda_{m}} x\Bigr)d x}{\displaystyle\int_{0}^{1}x J_{0}^{2}\Bigl(\sqrt{\lambda_{m}} x\Bigr)d x}, (24)
which determines the coefficients in the series (22).