Problem $221112$d

$($a$)$ Find VOUT as a function of time, i$.$e$.$ v$($t$),$ 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$)$ You know that I$($s$),$ the Laplace transform of i$($t$),=$ V$($s$)/$ Z$($s$).$ Right? So you can

find the current in the circuit by finding the current in the capacitor. Now that you

have V$($s$),$ i$.$e$.$ Vout$($s$),$ divide the expression of V$($s$)$ 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 i$($t$)$