= Free parallel RLC voltage decay
{title2=$V''+V'/(RC)+V/(LC)=0$}
With no external current, <Kirchhoff's current law> is $C\dot V+V/R+I_L=0$, and $L\dot I_L=V$. Differentiating gives the title equation. For $4R^2>L/C$, its real solutions are $Ae^{-t/(2RC)}\cos(\omega_dt+\phi)$ with $\omega_d^2=1/(LC)-1/(4R^2C^2)$. The original current balance fixes the compatible initial <inductor> current. A single <complex exponential> represents a physical <voltage> through its <real part>; general complex solutions require both conjugate characteristic roots.
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