Transients question - Understanding the current
Very simple.

You'll have a current through sw2 due to the 12V battery limited by the 25 Ohm resistor.

This the + 0.48 component (taking this as a positive current direction)

Then you have the decay of the inductor (current in the opposite direction hence the '-'sign)

The steady state current will be determined by the total resistance (25+15 = 40 ohms) of the circuit and the 12 volt battery  (i.e. 12/40 = 0.3 Amps)

and finally the time constant for the exponential decay = T = L/R = 5/15 
 giving you -3t (= -t/T)) as the exponent. 

Put them all together to give 0.48 -0.3e^-3t