Transient analysis with current source
Your initial conditions are correct. There is no short circuit in here. Take a look at the circuit. We have 2 time intervals:

@t&lt;0:


the switch is open. without a return path back to its source, there
can be no current
You essentially have a circuit with NO sources; the current source is
an open circuit


@t--> infinity:
circuit is in steady state:


this means the capacitor is open and the inductor is shorted into a
wire
By definition, a wire has no potential difference.
Since there is an open, the resistor behind the capacitor receives no
current, and has zero volts (ohm's law).
The capacitor itself is fully charged to maximum voltage, but is not
  considered in kvl analysis because it is open.
In essence, you have a single Kvl loop with inductor, current source and 
  resistor. ==> I=3, VL=0, Vr=4*3=12v.
Since the resistor is in parallel with the capacitor, they share the same 
  voltage: 12v.


From here, you can set up the second order equations need to solve for the transient responses, so I'll leave it to you. 
Hint: decide what type of RLC circuit it is, whether it's critically, under, or overdamped, and go from there.