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

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

Note $2$: Notice that I defined the initial condition for the inductor current even thought it's zero. I did that because my experience with LTspice is that if you assign one initial condition $($e$.$g$.$ VOUT $\backslash (=-1\backslash$mathrm$\{$~V$\}\backslash )),$ then you need to specify them all, even the ones that are zero. Otherwise, LTspice makes some other decision for you and you don't get what you expect.

Note $3$: Once you've determined the Laplace domain $($s$-$domain$)$ representations of the components, redraw the circuit with those s$-$domain representations shown.