The switch in this circuit has been closed for a long time, and is opened at time t$=0.$ Suppose IS$=18$mA$,$ L$=26$mH$,$ C$=0\mathrm{.}6$F$,$ and the resistance values are RS$=300$ and R$=390.$

$($a$)$ Determine the following initial conditions for the inductor current iL:

iL$(0)=$ mA

diLdt

$(0+)=$ mA$/$s

$($b$)$ Determine the roots of the characteristic equation for iL$.$

$($If the roots are real, state the smaller$-$magnitude one first; if they are complex, state the one with positive imaginary part first.$)$

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

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

$($c$)$ Determine the inductor current iL$($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:

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

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

a $=$ s$-1$

b $=$ s$-1$

k$1=$ mA

k$2=$ mA