Chapter 6.2, Problem 10P

In section $4.2$ it was shown that if ${\text{x}}_{\text{1}}$ and ${\text{x}}_2$ are solutions of

$\text{x}\text{'}=\left(\begin{array}{cc}{p}_{11}\left(t\right)& p_{12}\left(t\right)\\ p_{21}\left(t\right)& p_{22}\left(t\right)\end{array}\right)\text{x}$ ,

on an interval $I$, then the Wronskian $W$ of ${\text{x}}_{\text{1}}$ and ${\text{x}}_2$ satisfies the differential equation $W\text{'}=\left(p_{11}+p_{22}\right)W$. A generalization of that proof shows that if ${\text{x}}_{\text{1}}\text{,}\mathrm{.......}{\text{,x}}_{\text{n}}$ are solutions of Eq.(4) on $I$, then the Wronskian of ${\text{x}}_{\text{1}}\text{,}\mathrm{.......}{\text{,x}}_{\text{n}}$, denoted by $W$, satisfies the differential equation

$W\text{'}=\left(p_{11}+p_{12}+\mathrm{.....}+p_{nn}\right)W=tr\left(\text{P}\left(t\right)\right)W.$         (i)

(a) Explain why Eq(i) also implies that $W$ is either identically zero or else never vanishes on $I$ in accordance with Theorem 6.2.5.

(b) If ${\text{y}}_{\text{1}}\text{,}\mathrm{.......}{\text{,y}}_{\text{n}}$ are solutions of Eq(28), find a counterpart to Eq(i) satisfied by $W=W\left[{\text{y}}_{\text{1}}\text{,}\mathrm{........}{\text{,y}}_n\right]\left(t\right)$.