AC nodal analysis with multiple voltage sources at same frequency

[R0CC0]

Homework Statement

I've attached the question.

So I'm not sure which method should be used to solve this. I was thinking superposition, but the voltage supplies are are both at the same ω value.

Homework Equations

v(t) = V_maxcos(ωt + $\phi$)
Z_L = jωL
Z_C = 1/jωL

The Attempt at a Solution

convert everything to phasor form
V_s1 = 20$\angle$0°
V_s2 = 20cos(1000t - 90°) = 20$\angle$-90°
10mH = j10 = 10$\angle$90°
0.1mF = -j10 = 10$\angle$-90°

Then I'm not really sure what to do next, do I use superposition or mesh analysis or etc. and then how does that work with phasors?
So would appreciate a push in the right direction.

 

[NascentOxygen]

Since you have marked all node voltages, you could use: current through the inductor + current through the capacitor = current in the resistor
and with only one unknown, solve this to find V_x

 

[rude man]

In terms of phasors, you can let Vs2 = 20, then Vs1 = 20e^jπ/2.

You can then solve using kvl or you can use superposition to solve for one of the sources at a time, then add the results.

Superposition is allowable if the sources are independent, which here they are.

Had the frequencies been different you would have had to use superposition, solving two separate problems individually, with different complex impedances for each problem. Superpositoon would have to be done in the time domain.