The switch in this circuit has been open for a long time, and capacitor C$2$ is initially uncharged when the switch is closed at time t$=0.$ Suppose VB$=10$V$,$ C$1=1\mathrm{.}4$F$,$ C$2=5$F$,$ and the resistance values are Rs$=360$k$$ and R$=340$k$.$

$($a$)$ Apply the Laplace transform to this circuit for times t$>0,$ to determine the voltage drop across capacitor C$2$ in the s$-$domain, when it is written in the form

V$2($s$)=$

a$1$ s $+$ a$0$s$($s$2+$ b$1$ s $+$ b$0)$

$.$

a$1=$ V$/$s

a$0=$ V$/$s$2$

b$1=$ s$-1$

b$0=$ s$-2$

$($b$)$ Determine the non$-$zero poles of V$2($s$).($If they are real, list the one with smaller magnitude first; if they are complex, list the one with positive imaginary part first.$)$

s$1=+$ j $$ s$-1$

s$2=+$ j $$ s$-1$

$($c$)$ Determine the voltage drop v$2($t$)$ as a function of time t $($in seconds$),$ for t$>0.$ Identify the numerical values of the parameters in whichever of the following forms is appropriate:

v$2($t$)=$ k$1$ e$$at $+$ k$2$ e$$bt $+$ k$3($with $|$a$|<|$b$|),$ or

v$2($t$)=$ e$$at $[$k$1$cos$($bt$)+$ k$2$sin$($bt$)]+$ k$3.$

a $=$ s$-1$

b $=$ s$-1$

k$1=$ V

k$2=$ V

k$3=$ V