= Solution
Set $v=xu$. Direct <differentiation> gives $u''+2u'/x=v''/x$, so $-v''=\lambda v$. Finiteness of $u$ at zero requires $v(0)=0$, and the other endpoint gives $v(\pi/2)=0$. The <Dirichlet eigenvalues> are positive: <integration by parts> gives $\lambda\int|v|^2=\int|v'|^2$, and a nonzero $v$ cannot have zero right-hand side. Thus $v=C\sin(\sqrt\lambda x)$, and the second endpoint gives
$$
\boxed{\lambda_n=4n^2,\qquad u_n(x)=\frac{\sin(2nx)}x,\qquad n=1,2,\ldots.}
$$
Define $u_n(0)=2n$ by its <limit>. This is the <radial Dirichlet spectrum in three dimensions> with radius $\pi/2$. Its weighted squared <norm> is $\int_0^{\pi/2}x^2u_n^2dx=\pi/4$.
Back to article page
