# Discretize using a forward-Euler scheme

## Homework Statement

Consider the differential equation

y'''-y''=u

Discretize (1) using a forward-Euler scheme with sampling period

\Delta=1

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).

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BvU
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. $${y_k \over u_k}\ = \ {1\over (z−1)^3 − (z−1)^2 } {\rm\quad ?}$$