If the circuit has been running with the switch 'off' for a long time and then suddenly the switch is closed, there is still energy in the inductor (1/2)*L*i^2 where i is the initial current level, so this basically turns into a circuit with an inductor in parallel with a capacitor and in parallel with a resistor, with initial energy in the inductor only. So it's a parallel RLC circuit with initial conditions iL=i(0) and vC=0.

One way to solve is you can place an initial current generator in parallel to the inductor and then call the inductor current zero. The initial current generator has the value of the initial current just before the switch is turned on.

So it turns into a regular parallel RLC circuit driven with a current source at t=0.

Now to get exactly 0.12 volts DC across the capacitor constantly is impossible because eventually all the energy will be eaten up by the resistance. To force this condition you would need some additional external energy input such as another voltage source or instead of closing the switch completely use a switch with some intrinsic resistance instead. That will keep some energy pouring into the circuit, and then solving the DC solution will tell you what resistance is required in the switch to keep the cap voltage at 0.12v or whatever you need.

If this was to be 0.12v peak AC then you would have to get rid of all the resistance and solve for the steady state voltage across the cap but that would mean switching out ALL of the resistance so as to form a natural oscillator which would mostly only be practical in theory or maybe in a good laboratory.