Cancelling transient state in RLC circuit
Here is one way. The steady state forced response of the voltage across the capacitor would be:
$$ v_C(t) = K\sin(wt + \theta) $$
$$ i_C(t) = C \frac{dv_C}{dt} = CwK\cos(wt + \theta)$$

At time = 0, apply the two initial conditions and equate them to the pure steady state forced response:
$$ v_C(0) = 0 = K\sin(\theta) $$
$$ i_C(0) = i_L(0) = 0 = CwK\cos(\theta) $$

It is clear that such a solution does not exist because it is not possible for:
\$ \sin(\theta) = \cos(\theta) = 0 \$ for any \$\theta\$.

Therefore, at time 0, the response cannot be pure steady state. There must be terms for natural/transient response in those equations at time 0.

A summary of this is that the steady state response of the given forcing function is sinusoidal, so the voltage and current across the capacitor has a phase difference of 90\$^\circ\$. But at time zero, the initial conditions dictate the voltage and current across the capacitor to be 0. Therefore, there is no non-trivial steady state only solution at time 0.