Write the differential equation for an $LC$ circuit.

(N/A) In an $LC$ circuit consisting of an inductor $L$ and a capacitor $C$,the sum of the potential drops across the inductor and the capacitor must be zero according to Kirchhoff's voltage law.
Let $q$ be the charge on the capacitor at any time $t$.
The potential difference across the capacitor is $V_C = \frac{q}{C}$.
The potential difference across the inductor is $V_L = L \frac{di}{dt}$.
Since $i = \frac{dq}{dt}$,the current is the rate of change of charge.
Thus,$V_L = L \frac{d^2q}{dt^2}$.
Applying Kirchhoff's loop rule: $V_L + V_C = 0$.
Substituting the expressions: $L \frac{d^2q}{dt^2} + \frac{q}{C} = 0$.
This is the differential equation for the $LC$ circuit,which can also be written as $\frac{d^2q}{dt^2} + \frac{1}{LC} q = 0$.