$4\mathrm{.}3.$ In the network given, the initial voltage on $C_1$ is $V_1$ and on $C_2$ is $V_2$ such that $v_1(0)=V_1$ and $v_2(0)=V_2.$ At $t=0,$ the switch $K$ is closed.

$($a$)$ Find $i(t)$ for all time. $($b$)$ Find $r_1(t)$ for $t>0.($c$)$ Find $r_2(t)$ for $t>0.($d$)$ From your results on $($b$)$ and $($c$),$ show that $r_1()=r_2().$

$($e$)$ For the following values of the elements, $R=1,C_1=1F,C_2=\frac{1}{2}F,V_1=2V,V_2=1V,$ sketch $i(t)$ and $r_2(t)$ and identify the time constant of each.

Fig. P$4-3.$