Problem $221112$d

$($a$)$ Find VOUT as a function of time, i$.$e$.\backslash (\backslash$mathrm$\{$v$\}(\backslash$mathrm$\{$t$\})\backslash ),$ using the Laplace transform approach.

Note $1$: Current for the inductor is defined from left to right. I think you know that that's my normal convention, but I$\text{'}$m stating explicitly just to preclude any confusion.

Note $2$: Once you've determined the Laplace domain $($s$-$domain$)$ representations of the components, redraw the circuit with those s$-$domain representations shown.

$($b$)$ Simulate the circuit and your answer to part $($a$).$ Plot the responses together and give me a screenshot.

$($c$)$ You know that $\backslash (\backslash$mathrm$\{$I$\}(\backslash$mathrm$\{$s$\})\backslash ),$ the Laplace transform of $\backslash (\backslash$mathrm$\{$i$\}(\backslash$mathrm$\{$t$\}),=\backslash$mathrm$\{$V$\}(\backslash$mathrm$\{$s$\})/\backslash$mathrm$\{$Z$\}(\backslash$mathrm$\{$s$\})\backslash ).$ Right? So you can find the current in the circuit by finding the current in the capacitor. Now that you have $\backslash (\backslash$mathrm$\{$V$\}(\backslash$mathrm$\{$s$\})\backslash ),$ i$.$e$.$ Vout$($s$),$ divide the expression of $\backslash (\backslash$mathrm$\{$V$\}(\backslash$mathrm$\{$s$\})\backslash )$ that you put into the residue command by the impedance of the capacitor to get an expression for I$($s$).$ Transform that expression back to the time domain as $\backslash ($ i$($t$)\backslash ).$