Differential Eqn. of a 3rd order Filter

[Jaded1]

Hi guys :) I'm going through some prep. work for my modules in Fall and I've come across a question I'm having some difficulty with.

Homework Statement

The question has a transfer function for which I need to find the differential equation relating the output voltage and the input voltage.

Homework Equations

Transfer function: H(s) = 15/(s³ + 6s² + 15s + 15)

The Attempt at a Solution

I created the above circuit using a 3rd order sallen-key lowpass filter as in this website:http://sim.okawa-denshi.jp/en/Sallenkey3Lowkeisan.htm. I then split the circuit into 3 parts - A,B,C.

I then deduced 3 voltage equations for the 3 nodes using kirchhoffs circuit laws as follows;

Va = (Va-Vin)/Z1 + (Va-Vb)/R2 + (Va-Vout)/Z2 = 0

Vb = (Vb-Va)/R2 + (Vb-Vout)/Z2 + (Vb-Vc)/R3 = 0

Vc = (Vc-Vb)/R3 + (Vc-0)/Z3 = 0

where Vc = Vout.

Are the above equations correct? I'm not very comfortable with 3rd order circuits as of yet, and am not sure if I got the equations right. I've gotten to this point, but after this I'm not sure what I need to do. If I simply the above equations I will get the transfer function, but that's not what I want to find, how do I proceed from here?

 

[rude man]

If you're trying to derive the diff. eq. for the given transfer function, what are you doing with Sallen-Key filters?

A transfer function is readily transposed to a diff. eq. since s → d/dt. Take it from there ...