How does the voltage drop to negative values at t = 0 when the voltage starts with 0 V?
Figure 1. The current after the switch opens.

You can think of an inductor as liking to maintain the current through it - in the short-term at least. The result of this is that at the moment you open the switch that current reverses through the 5k resistor.

Now observe that since current is flowing clockwise around the loop and the polarity of the voltage across the resistors will be as shown.

Finally, since the bottom of the inductor and 5k resistor are tied to GND and the right side of the 15k is negative then to top of the inductor must be negative too.