Current in Buck Converter
An inductor has inductive properties plus some DC resistance, but by itself it is a linear device, much like a resistor is a linear device by itself. The current through the inductor is linear because the voltage is either fully ON or OFF at the mosfet switch. The inductor core can only magnetize or demagnetize at a fixed maximum rate. Only a non-linear device such as a diode or transistor (bjt) would have non-linear behavior by themselves. A non-linear rise or fall time is accomplished by combining resistors and capacitors and/or inductors together to form a time-constant. This does not include circuits with constant-current sources or sinks, as they would intentionally create a linear rise and fall time. Notice that in your equations you did not insert any non-linear properties. There are many post on this site and websites involving "Switch mode power supplies" that offer books of data and pages of math to help in understanding the unique but predictable properties of inductors.