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## 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!

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