Problem $\backslash$#$2(30$ points$)$

In this circuit, the switch has been open for a long time and at $\backslash (\backslash$mathrm$\{$t$\}=0\backslash )$ it closes. It is given that at $\backslash (\backslash$mathrm$\{$t$\}=0\backslash )-$ there is no energy stored in the inductor or the capacitor.

a$)$ Show that when the switch is closed, the current $\backslash ($ I$(\backslash$mathrm$\{$~s$\})\backslash )$ in the s$-$domain is given by:

b$)$ Perform an inverse Laplace transform and determine the current $\backslash ($ i$(\backslash$mathrm$\{$t$\})\backslash )$ in the time$-$domain for $\backslash (\backslash$mathrm$\{$t$\}\backslash$geq $0\backslash ).$

c$)$ Plot the current $\backslash ($ i$(\backslash$mathrm$\{$t$\})\backslash )$ from $\backslash (\backslash$mathrm$\{$t$\}=0\backslash )$ until a time at which the current reaches a steadystate value.

d$)$ On the plot of part $($c$),$ indicate the transient and steady state periods of time.

e$)$ Using the known behavior of the inductor and capacitor explain the steady state value of the current.