= Solution
The homogeneous <characteristic equation> is $r^2-2r+2=0$, with roots $1\pm i$. Choose the endpoint-adapted homogeneous solutions
$$
u(x)=e^x\sin x,\qquad v(x)=e^x\cos x,
$$
so that $u(0)=0$ and $v(\pi/2)=0$. Their <Wronskian> is
$$
W(x)=u(x)v'(x)-u'(x)v(x)=-e^{2x},
$$
which never vanishes. A continuous <Green function> with the required endpoints has the form $C(\xi)u(x_<)v(x_>)$, where $x_<=\min(x,\xi)$ and $x_>=\max(x,\xi)$. Its <derivative> jump is $C(\xi)W(\xi)$. Because the leading coefficient is one, part (b) requires this to equal one. Therefore
$$
\boxed{G(x;\xi)=-e^{x-\xi}\sin x_<\cos x_>.}
$$
In particular, the branches are $-e^{x-\xi}\sin x\cos\xi$ for $x\leq\xi$ and $-e^{x-\xi}\sin\xi\cos x$ for $x\geq\xi$. This is the <Dirichlet Green function for a conjugated harmonic oscillator> with interval length $\pi/2$, drift parameter one and oscillator wave number one.
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