Boost converter: doubts on signs voltages
For the boost to be in steady-state the inductor current delta (ramp up) during the on time has to equal the inductor current delta (ramp down) in the off time.

Let's assume Vd is the input voltage and Vo the output voltage, and the diode is ideal.







simulate this circuit &ndash; Schematic created using CircuitLab

Rearrange your equation

$$Vd * tOn + (Vd - Vo) * tOff = 0$$

Like this:

$$Vd * tOn = (Vo - Vd) * tOff$$

Using the relation $$V=L*di/dt$$ and $$di = V*dt/L$$ you can see that the above equation balances the inductor up-slope during the on time with the down slope during the off time.  The L cancels out on both sides of the equation.

The waveforms look like this, (From Here)