= Solution
For <free parallel RLC voltage decay>, measure $V=V_A-V_B$ and branch currents from A to B. Then
$$
I_C=C\dot V,\qquad I_R=V/R,\qquad
I_L(t)=I_L(0)+\frac1L\int_0^tV(s)\,ds.
$$
There is no external current, so <Kirchhoff's current law> gives $I_C+I_R+I_L=0$. Differentiating yields
$$
\boxed{C\ddot V+\frac1R\dot V+\frac1LV=0.}
$$
For $V\propto e^{st}$ the <characteristic equation> is $Cs^2+s/R+1/L=0$. Its <underdamped> roots give
$$
\boxed{\gamma=\frac1{2RC},\qquad
\omega=\sqrt{\frac1{LC}-\frac1{4R^2C^2}}.}
$$
The supplied inequality makes $\omega$ real. A real physical <voltage> has the general form $V=Ae^{-\gamma t}\cos(\omega t+\phi)$, the <real part> of the printed complex expression. A general complex-valued solution uses both roots, $e^{-\gamma t}(C_+e^{i\omega t}+C_-e^{-i\omega t})$. The <inductor>'s initial current is fixed consistently by the undifferentiated current balance; it is not an arbitrary additional voltage-mode constant.
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