How do I determine the unknowns for this equation?
If \$K_1 = 0\$ and steady-state current is \$i_{ss}=0\$:
$$ i(t) = K_2e^{-\alpha t}sin(\omega t)$$

Then,
$$ \frac{di(t)}{dt} = K_2 \left[ -\alpha e^{-\alpha t}sin(\omega t) + \omega cos(\omega t)e^{-\alpha t} \right ]$$

$$ K_2 = \frac{1}{\omega}\frac{di(0_+)}{dt}  $$

$$ \frac{di(0_+)}{dt} = \frac{v_{L_{(0+)}}}{L}$$

But \$ v_{L_{(0+)}} = 12 V \$ and \$L=0.1 H\$
and
$$K_2 = 0.12 A$$