Chapter 11.RP, Problem 8RP

Find a formal eigenfunction expansion for the solution to the given nonhomogeneous boundary value problem where $f$ is a continuous function and $\mu$ is not an eigenvalue of the corresponding homogeneous problem.

a. ${\left(x{y}^{\prime }\right)}^{\prime }-\left(49/x\right)y+\mu xy=f\left(x\right)$, $0<x<1$,

$y\left(x\right)$, $y^{\prime }\left(x\right)$ remain bounded as $x\to 0^{+}$, $y\left(1\right)=0$

b. ${\left[\left(1-x^2\right)y^{\prime }\right]}^{\prime }+\mu y=f\left(x\right)$, $-1<x<1$,

$y\left(x\right)$, $y^{\prime }\left(x\right)$ remain bounded as $x\to \pm 1$