About SPICE: Should I use transient analysis or DC linear analysis?
You would be best to use a transient analysis. Rather than using the time parameter you could setup the current source with the pulse attribute, then specify the rise/fall time accordingly if using a standard current source (Ix). Or use an arbitrary source (Bx) and express the signal mathematically using time as a parameter.  

For example, here is the netlist for your circuit in LTSPice using an arbitrarily behavioural current source:  

* C:\Program Files\LTC\LTspiceIV\current rc.asc
R1 N002 N001 2
C1 N002 N001 50mF
R2 N002 0 3
C2 N002 0 20mF
B1 0 N001 I=time
.tran 0 15 1m uic
.backanno
.end


Here is the simulation with V1 - V2 plotted: