Question about AC and DC on a RC circuit
It looks like you are mixing transient response analysis (the exponential equation) and steady-state ac analysis (the j\$\omega\$ part). In a situation like this, which is neither transient nor steady-state, I think you go back to basics:
$$i_c = C \frac{dv_c}{dt}$$
and
$$v_c = \cfrac{1}{C}\int i_c\, {dt} + v_c(0)$$

You also know that \$i_c = i_r\$ and \$V_c = V_s - R\,\cdot i_R\$. Looks like some math ahead.