Inductor induced voltage understanding
Hopefully, this will explain how there are two initial conditions. The first one is obvious (the current taken by the coil when the transistor was conducting) and the initial condition of di/dt when the transistor open-circuited.

When the transistor open-circuits the static initial condition for the inductor is "so-many amps" flowing. That intial condition will generate a large voltage (theoretically and simplistically infinite) but we know parasitic (or intentional clamp) components restrict us to hundreds of volts.

Following the open-circuit event and making the simplistic assumption that the voltage stays constant for some time afterwards (assume there was a zener diode clamp at 100 volts for instance), we know that di/dt = V/L.

So we know the static initial condition (let's say 100 mA), we know V (100 volts) and we hopefully know the inductance (say 100 mH). To maintain this situation current falls from 100 mA towards zero at a rate of 100 volts / 100 mH = 1000 amps per second or 1 A per millisecond.

This means that after 100 micro seconds the current hits zero and there is no energy left in the inductor - it's all burnt off by the zener. At this point the voltage instantly collapses to zero volts.


  I know it's weird but please, do not consider the Protection Diode
  across it


Without a protection diode and in the absense of parasitic capacitance in the transistor or inductor then the voltage is of course infinite and the event lasts for zero time. I don't find that to be a helpful scenario but you asked for it!