Chapter 11.3, Problem 23E

In Problems 17 -24, (a) determine the normalized Eigenfunctions for the given problem and (b) use the results of part (a) to express $f\left(x\right)=x$ in an Eigen function expansion. (These are the same equations as in Problems 13 through 20 of Exercises 11.2 .)

${\left(x{y}^{\prime }\right)}^{\prime }+\lambda x^{-1}y=0$,

$y^{\prime }\left(1\right)=0,y\left(e^{\pi }\right)=0$