Chapter 11.5, Problem 15E

Derive the solution to Problem $12$ given in equation $17$.

12. $x{y}^{″}+y^{\prime }+x^{-1}y=f$;

$y\left(1\right)=0$, $y\left(e^{\text{π}}\right)=0$

$\begin{array}{l}\varphi \left(x\right)=\{\begin{array}{l}\frac{2}{\pi },0\le x\le \pi /2\\ 1-\frac{2}{\pi },\pi /2\le x\le \pi \end{array}\\ \end{array}$ (17)