using mesh analysis with capacitors and impedances
Analysis with impedances assumes a sinusoidal steady-state operation. Since the voltages in your circuit will not change direction unless you switch the switches (thus changing the equivalent circuit), this analysis is not appropriate.

You'll need to analyze the transient behavior of your circuit, not the sinusoidal steady state behavior. You can do this with mesh analysis, nodal analysis, or an ad hoc combination of the two.