Chapter 4.7, Problem 32E

By completing the following steps, prove that the Wronskian of any two solutions $y_1$, $y_2$ to the equation $y^{″}+p{y}^{\prime }+qy=0$ on $\left(a,b\right)$ is given by Abel’s formula

$W\left[y_1,y_2\right]\left(t\right)=C\exp \left\{-{\displaystyle \underset{t_0}{\overset{t}{\int }}p\left(\tau \right)d\tau }\right\}$,

$t_0$ and $t$ in $\left(a,b\right)$,

where the constant $C$ depends on $y_1$ and $y_2$.

a. Show that the Wronskian $W$ satisfies the equation $W^{\prime }+pW=0$.

b. Solve the separable equation in part (a).

c. How does Abel’s formula clarify the fact that the Wronskian is either identically zero or never zero on $\left(a,b\right)$?