Chapter 7.6, Problem 20E

(a)

To determine

To show: The system of differential equation for charge on the capacitor $q\left(t\right)$ and the current $i_3\left(t\right)$ in the electrical network is $R_1\frac{dq}{dt}+\frac{1}{C}q+R_1{i}_3=E\left(t\right)$ and $L\frac{d{i}_3}{dt}+R_2{i}_3-\frac{1}{C}q=0$ using Laplace transformation.

(b)

To determine

The charge on the capacitor if $L=1\text{h, }{R}_1=1\Omega ,R_2=1\Omega ,C=1\text{f}$ and $i_3\left(0\right)=0,q\left(0\right)=0,E\left(t\right)=\{\begin{array}{l}0,0<t<1\\ 50e^{-t},t\ge 1\end{array}$ using Laplace transformation.