Past exam of the mathematical methods I course of the University of Oxford / 2022 / 6

Let $a=R/(2L)$ and ${\omega }_d=\sqrt{1/(LC)-a^2}>0$. The roots of the characteristic polynomial are $-a\pm i{\omega }_d$, so the real underdamped solution is $Q=e^{-at}(A\cos {\omega }_{d}t+B\sin {\omega }_{d}t)$. The initial charge gives $A=Q_0$ and the zero initial current gives $B=a{Q}_0/{\omega }_d$. HenceThis solves the homogeneous RLC circuit equation supplied in the question. A nonzero applied voltage would appear as a forcing term and require a particular solution; it is not present in that equation.