Chapter 5.2, Problem 22P

In section $4.1$ the differential equation for the charge on the capacitor of the RCL circuit illustrated in Figure $\mathrm{5.2.1}$ was shown to be

$Lq\text{'}\text{'}+Rq\text{'}+\frac{1}{C}q=e\left(t\right)$         (iii)

Take the Laplace transform of Eq. (iii) to derive Eq. (12) for $Q\left(s\right)$ and then use Eq. (9) to derive Eq. (13) for $I\left(s\right)$.

Eq. (12): $Q\left(s\right)=\frac{\left(Ls+R\right)q\left(0\right)+Li\left(0\right)+E\left(s\right)}{L{s}^2+sR+\frac{1}{C}}$

Eq.(9): $I\left(s\right)=sQ\left(s\right)-q\left(0\right)$

Eq.(13): $I\left(s\right)=\frac{Lsi\left(0\right)+sE\left(s\right)-\frac{q\left(0\right)}{C}}{L{s}^2+sR+\frac{1}{C}}$