Should I use superposition if there is a sine wave current source and a DC voltage source in a circuit?

Question

^{simulate this circuit – Schematic created using CircuitLab}

The DC voltage source has an angular frquency of 0 rad/sec.

Do I need to:

1. short circuit the DC voltage source
2. calculate I(t) in time domain
3. open circuit the AC current source
4. calculate I(t)
5. And finally, add the two together

Or can I just do a Kirchoff Voltage Law in the right loop? $$(1.6-0.8j)I+12+(8+4j)I=0$$

This is the original circuit before I simplified the impedance:

Answer 1

Start off with a DC analysis (Open circuit the current source, and the caps, inductors are short circuits), assuming that resistor in series with the voltage source is 2 ohms (Difficult to read), then with both capacitors open circuit and the inductor short circuiting the 8 ohm resistor you get 12V across 4 + 2 ohms = 6 ohms = 2A for the DC terms of I(t).

Now, being careful of the fact that 2 radians/s is not the same thing as 2Hz, compute the AC impedances at the frequency of the current source and having replaced the DC voltage source with a short circuit solve for the AC component of I(t), then superposition gets you the answer.