Chapter 11.7, Problem 11E

To determine

(a)

To obtain:

The equation given as,

$-{\displaystyle {\int }_0^{1}x{\left[{\varphi }^{\prime }\left(x\right)\right]}^{2}dx}-v^2{\displaystyle {\int }_0^1{x}^{-1}{\left[{\varphi }^{\prime }\left(x\right)\right]}^{2}dx}+\lambda {\displaystyle {\int }_0^{1}x{\left[\varphi \left(x\right)\right]}^{2}dx}=0$

To determine

(b)

To show:

That for $v>0$ in the equation $(3)$ in part (a), the eigenvalue is $\lambda >0$

To determine

(c)

To show:

That for $v=0$ in the equation $(3)$ in part (a), the eigenvalue is $\lambda \ge 0$