# How to find the coefficients of the following differential equation

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)?

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

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 $$H_1(s)$$.