Find the charge in the steady state. What does steady state mean in this problem?

Question

A voltage $E(t)$ is applied to a RCL Circuit connected in series. The charge on the capacitor at any time, $t$ in seconds is given by :

$$q’’ + 9 q’ + 14 q =E(t) = \frac{1}{2} \sin t $$

Given that there is no initial charge, but an initial current of $1$ amperem what is the charge in the steady state ?

First, I solved this differential equation using Laplace transform and got

$$q(t) = \frac{1}{500} (110 e^{-7t} - 101e^{-7t} + 13 \sin t - 9 \cos t)$$

What does steady state mean ?

Why is the answer for the charge in steady charge $ q(t) = \frac{1}{500} (13 \sin t - 9 \cos t )$ (Without the part that I’ve taken out.)

Answer 1

This is a somewhat loose use of the term “steady state”. It's used here to mean that the transient exponentials have decayed and only the permanent oscillation remains. I'd rather call it a “quasi-steady state”. It's not steady if you focus on the time domain, where the quantities are still changing, but when dealing with electrical circuits, one often focuses on the frequency domain, and with that in mind, a single sine wave with a fixed frequency and amplitude is something like a steady state.