 Free parallel RLC voltage decay

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 $A{e}^{-t/(2RC)}\cos ({\omega }_{d}t+\varphi )$ with ${\omega }_d^2=1/(LC)-1/(4R^2{C}^2)$. 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.