Alternate derivation of output capacitor ripple voltage buck

[Dextrine]

Homework Statement

I'm trying to find the output voltage ripple of a buck converter by using just the waveforms and the inductor ripple.

Homework Equations

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I know that the answer is supposed to be D(1-D)Vin/(16*L*C*f^2) but I keep getting D(1-D)Vin/(8*L*C*f^2)The equations I end up using are Ic=C*dVc/dt and I let Ic=deltaIL.

I know deltaIL= D(1-D)Vin/(2*L*f) and I know that the period that the capacitor has increasing voltage is Ts/4

The Attempt at a Solution

with this knowledge, multiplying everything together gives D(1-D)Vin/(8*L*C*f^2)

I think I'm making some sort of mistake letting Ic=deltaIL and it makes be off by a factor of 2.Any help is very appreciated (this isn't homework, just my own curiosity. I've seen the derivation for capacitor
voltage ripple using the charge on the capacitor but I wanted to try to do it this way.)

 

[Baluncore]

Please provide a circuit diagram that identifies the components, currents and voltages.
Please sketch the waveforms you expect and show the voltage measurement points on the waves.