Discretize using a forward-Euler scheme

[Linder88]

Homework Statement

Consider the differential equation
\begin{equation}
y'''-y''=u
\end{equation}
Discretize (1) using a forward-Euler scheme with sampling period
\begin{equation}
\Delta=1
\end{equation}
and find the transfer function between u(k) and y(k)

Homework Equations

The Euler method is
$$
y_{n+1}=y_n+hf(x_n,y_n)
$$

The Attempt at a Solution

Laplace transform of (1) yields
$$
s^3Y(s)-s^2Y(s)=U(s)
$$
From my teacher I know that
$$
s=\frac{z-1}{\Delta}
$$
Using this formula on the Laplace transform of (1) yields
$$
\bigg(\frac{z-1}{\Delta}\bigg)^3y_{k}-\bigg(\frac{z-1}{\Delta}\bigg)^2{y_k}=u_k
$$
Substituting (2) in this equation yields
$$
(z-1)^3y_k-(z-1)^2y_k=u_k
$$
$$
y_{k+3}-y_{k+2}=u_k
$$
Now I want to find the transfer function between u(k) and y(k) but I don't see and y(k).
Can somebody please help me? I have my exam tomorrow!

 

[BvU]

I can't follow the step

Substituting (2) in this equation yields $$(z−1)^3 y_k − (z−1)^2 y_k =u_k $$

to $$
y_{k+3}-y_{k+2}=u_k $$Could you explain why this doesn't work out to e.g. $${y_k \over u_k}\ = \ {1\over (z−1)^3 − (z−1)^2 } {\rm\quad ?} $$