In this problem we show how to generalize Theorem 3.3 .2 (Abel's theorem) to
higher order equations. We first outline the procedure for the third order
equation
$$
y^{\prime \prime \prime}+p_{1}(t) y^{\prime \prime}+p_{2}(t)
y^{\prime}+p_{3}(t) y=0
$$
Let \(y_{1}, y_{2},\) and \(y_{3}\) be solutions of this equation on an interval
\(I\)
(a) If \(W=W\left(y_{1}, y_{2}, y_{3}\right),\) show that
$$
W^{\prime}=\left|\begin{array}{ccc}{y_{1}} & {y_{2}} & {y_{3}} \\\
{y_{1}^{\prime}} & {y_{2}^{\prime}} & {y_{3}^{\prime}} \\ {y_{1}^{\prime
\prime \prime}} & {y_{2}^{\prime \prime \prime}} & {y_{3}^{\prime \prime
\prime}}\end{array}\right|
$$
Hint: The derivative of a 3 -by-3 determinant is the sum of three 3 -by-3
determinants obtained by differentiating the first, second, and third rows,
respectively.
(b) Substitute for \(y_{1}^{\prime \prime \prime}, y_{2}^{\prime \prime
\prime},\) and \(y_{3}^{\prime \prime \prime}\) from the differential equation;
multiply the first row by
\(p_{3},\) the second row by \(p_{2},\) and add these to the last row to obtain
$$
W^{\prime}=-p_{1}(t) W
$$
(c) Show that
$$
W\left(y_{1}, y_{2}, y_{3}\right)(t)=c \exp \left[-\int p_{1}(t) d t\right]
$$
It follows that \(W\) is either always zero or nowhere zero on \(I .\)
(d) Generalize this argument to the \(n\) th order equation
$$
y^{(n)}+p_{1}(t) y^{(n-1)}+\cdots+p_{n}(t) y=0
$$
with solutions \(y_{1}, \ldots, y_{n} .\) That is, establish Abel's formula,
$$
W\left(y_{1}, \ldots, y_{n}\right)(l)=c \exp \left[-\int p_{1}(t) d t\right]
$$
for this case.