Suppose the equation
$$
P_{0}(x) y^{(n)}+P_{1}(x) y^{(n-1)}+\cdots+P_{n}(x) y=F(x)
$$
is normal on an interval \((a, b) .\) Let \(\left\\{y_{1}, y_{2}, \ldots,
y_{n}\right\\}\) be a fundamental set of solutions of its complementary
equation on \((a, b),\) let \(W\) be the Wronskian of \(\left\\{y_{1}, y_{2},
\ldots, y_{n}\right\\},\) and let \(W_{j}\) be the determinant obtained by
deleting the last row and the \(j\) -th column of \(W\). Suppose \(x_{0}\) is in
\((a, b)\), let
$$
u_{j}(x)=(-1)^{(n-j)} \int_{x_{0}}^{x} \frac{F(t) W_{j}(t)}{P_{0}(t) W(t)} d
t, \quad 1 \leq j \leq n,
$$
and define
$$
y_{p}=u_{1} y_{1}+u_{2} y_{2}+\cdots+u_{n} y_{n}
$$
(a) Show that \(y_{p}\) is a solution of \((\mathrm{A})\) and that
$$
y_{p}^{(r)}=u_{1} y_{1}^{(r)}+u_{2} y_{2}^{(r)} \cdots+u_{n} y_{n}^{(r)},
\quad 1 \leq r \leq n-1,
$$
and
$$
y_{p}^{(n)}=u_{1} y_{1}^{(n)}+u_{2} y_{2}^{(n)}+\cdots+u_{n}
y_{n}^{(n)}+\frac{F}{P_{0}}.
$$
HINT: See the derivation of the method of variation of parameters at the
beginning of the section.
(b) Show that \(y_{p}\) is the solution of the initial value problem
$$
\begin{array}{c}
P_{0}(x) y^{(n)}+P_{1}(x) y^{(n-1)}+\cdots+P_{n}(x) y=F(x) \\
y\left(x_{0}\right)=0, y^{\prime}\left(x_{0}\right)=0, \ldots, \quad
y^{(n-1)}\left(x_{0}\right)=0
\end{array}
$$
(c) Show that \(y_{p}\) can be written as
$$
y_{p}(x)=\int_{x_{0}}^{x} G(x, t) F(t) d t
$$
where
$$
G(x, t)=\frac{1}{P_{0}(t) W(t)}\left|\begin{array}{cccc}
y_{1}(t) & y_{2}(t) & \cdots & y_{n}(t) \\
y_{1}^{\prime}(t) & y_{2}^{\prime}(t) & \cdots & y_{n}^{\prime}(t) \\
\vdots & \vdots & \ddots & \vdots \\
y_{1}^{(n-2)}(t) & y_{2}^{(n-2)}(t) & \cdots & y_{n}^{(n-2)}(t) \\
y_{1}(x) & y_{2}(x) & \cdots & y_{n}(x)
\end{array}\right|,
$$
which is called the Green's function for (A).
(d) Show that
$$
\frac{\partial^{j} G(x, t)}{\partial x^{j}}=\frac{1}{P_{0}(t)
W(t)}\left|\begin{array}{cccc}
y_{1}(t) & y_{2}(t) & \cdots & y_{n}(t) \\
y_{1}^{\prime}(t) & y_{2}^{\prime}(t) & \cdots & y_{n}^{\prime}(t) \\
\vdots & \vdots & \ddots & \vdots \\
y_{1}^{(n-2)}(t) & y_{2}^{(n-2)}(t) & \cdots & y_{n}^{(n-2)}(t) \\
y_{1}^{(j)}(x) & y_{2}^{(j)}(x) & \cdots & y_{n}^{(j)}(x)
\end{array}\right|, \quad 0 \leq j \leq n .
$$
(e) Show that if \(a
