How to find the coefficients of the following differential equation

Question

An arbitrary signal $v(t)$ pass through the following system,

$w'(t) + 5 w(t) = v'''(t) + 320v''(t) + 40 v' (t) + 40v(t)$

How to determine the coefficients of the following differential equation, where the input signal is $w(t)$ and the output signal is $z(t)$, So that the output $z(t) = v(t)$ (after long time when there is no transient exist any longer)?

$z'''(t) + d_2 z''(t) + d_1 z'(t) + d_0 z(t) = e_3 w'''(t) + e_2 w''(t) + e_1 w'(t) + e_0 w(t)$

Answer 1

Since this a homework type problem, only a few hints that should help you solve this problem yourself:

1. determine the transfer function of the first system by transforming the given differential equation and calculating $H_1(s)=W(s)/V(s)$.
2. do the same with the second system to obtain $H_2(s)=Z(s)/W(s)$.
3. from $Z(s)=V(s)$, determine $H_2(s)$ in terms of $H_1(s)$.