= Solution
For each source point $z$, construct the <Green function> from homogeneous solutions on either side:
$$
G(x,z)=\begin{cases}A(z)y_a(x),&a\leq x<z,\\B(z)y_b(x),&z<x\leq b.\end{cases}
$$
The endpoint conditions are then automatic. The equation $L_xG=\delta(x-z)$ requires continuity of $G$ and a unit jump in its first <derivative>: a discontinuity in $G$ would introduce an unwanted delta <derivative>, while integrating across the source fixes $G_x(z^+,z)-G_x(z^-,z)=1$. Thus
$$
A y_a(z)=B y_b(z),\qquad B y_b'(z)-A y_a'(z)=1.
$$
Solving this two-by-two system, with the <Wronskian> $W=y_a y_b'-y_b y_a'\ne0$, gives
$$
\boxed{G(x,z)=\begin{cases}y_a(x)y_b(z)/W(z),&a\leq x\leq z,\\y_b(x)y_a(z)/W(z),&z\leq x\leq b.\end{cases}}
$$
This derivation also remains valid if one of the solution values at the matching point vanishes: only the <Wronskian> must be nonzero. For a regular nonsingular two-point problem it supplies the inverse kernel; at a singular endpoint, uniqueness requires separate checking, as part (d) illustrates.
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