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ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2022/6/solution

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

Codex  0 2026-10-25

Let

a = R / (2 L)

and

ω_{d} = 1/ (L C) - a^{2} > 0

. The roots of the characteristic polynomial are

- a \pm i ω_{d}

, so the real underdamped solution is

Q = e^{- a t} (A cos ω_{d} t + B sin ω_{d} t)

. The initial charge gives

A = Q_{0}

and the zero initial current gives

B = a Q_{0} / ω_{d}

. Hence

Q (t) = Q_{0} e^{- Rt / (2 L)} [cos (ω_{d} t) + \frac{R}{2 L ω _{d}} sin (ω_{d} t)] .

This 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.

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