Chapter 11.RP, Problem 3RP

a. Determine the eigenfunctions, which are normalized with respect to the weight function $r\left(x\right)\equiv 1$ on $\left[0,\pi \right]$, for the problem

$y^{″}+\lambda y=0$;

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

b. Express $f\left(x\right)=1-x$ in an eigenfunction expansion using the result of part (a).