I would like to find the solution of the below differential equation of a driven RLC:
$$L \frac{d^{2} Q}{d t^{2}}+R \frac{d Q}{d t}+\frac{Q}{C}=V_{0} \sin \omega t $$
I have solved the homogenuous case, and now, since I have a driving force, I want to find a partial solution. My guess is to try a $q_p(t)=Acos(wt)+Bsin(wt)$, then calculate first and second derivative, and substitute to the initial equation. After that, gather the sin terms together, and the cos terms together and set the multiplier of sin terms equal to V0, the multiplier of cos term equal to 0, and solve the system to find A, B.
I have found an analoguous procedure here, but everyone does calculations for the current, not the charge. Can this be done or am I wasting my time?
Furthermore, I would like some guidance about solving the above equation using Laplace transform. Does it have to do with the use of phasors as well?
Lastly, in resonance in mechanical driven harmonic oscillations, the resonant frequency is somewhat below the natural frequency of the system. Does the same occur with electrical driven harmonic oscillations?
W.r.t the Laplace transform technique, having in mind that
$$ \mathcal{L}(\dot q(t))=s\mathcal{L}(q(t))-q(0) = Q(s)-q(0) $$
and
$$ \mathcal{L}(\sin(\omega t)) = \frac{\omega}{s^2+\omega^2} $$
we have
$$ \left(s^2L+sR+\frac 1C\right)Q(s) = V_0\frac{\omega}{s^2+\omega^2}+L(\dot q(0)+s q(0))+R q(0) $$
or
$$ Q(s)= \frac{V_0\frac{\omega}{s^2+\omega^2}}{s^2L+sR+\frac 1C}+\frac{L(\dot q(0)+s q(0))+R q(0)}{s^2L+sR+\frac 1C} = Q_p(s) + Q_h(s) $$
where $Q_h(s)$ is the homogeneous response Laplace transform and $Q_p(s)$ is the particular due to the forcing input. We have also that $I(s) = s Q(s)$