State-Space Model of a Buck-Converter in DCM
There are three states each cycle:


Switch closed, your first equation applies (less the transistor drop).
Switch open, diode conducting; your second equation applies (less a diode drop, but I assume from your use of a IGBT that you consider a diode drop to be negligible).
Switch open, diode not conducting; in theory the coil has neither current nor voltage (in practice it'll be ringing like a bell), and the cap voltage is only affected by the output current


Your challenge is that the time that the converter stays in the switch open, diode conducting state depends on the coil current when the switch opens -- the switch-over happens when the circuit wants it to, not from any external control.