Existence of Fundamental Solution Sets. By Theorem
$1,$ for each $j=1,2, \ldots, n$ there is a unique solution
$y_{j}(x)$ to equation $(17)$ satisfying the initial conditions
$y_{j}^{(k)}\left(x_{0}\right)=\left\{\begin{array}{ll}{1,} & {\text { for } k=j-1} \\ {0,} & {\text { for } k \neq j-1, \quad 0 \leq k \leq n-1}\end{array}\right.$ .
(a) Show that $\left\{y_{1}, y_{2}, \ldots, y_{n}\right\}$ is a fundamental solution
set for $(17) .$ [Hint: Write out the Wronskian at $x_{0} . ]$
(b) For given initial values $\gamma_{0}, \gamma_{1}, \ldots, \gamma_{n-1},$ express
the solution $y(x)$ to $(17)$ satisfying $y^{(k)}\left(x_{0}\right)=\gamma_{k}$
$k=0, \ldots, n-1,[$ as in equations $(4)]$ in terms of
this fundamental solution set.