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Write the difference equations in Exercises 29 and 30 as first-order systems, $\mathbf{x}_{k+1}=A \mathbf{x}_{k},$ for all $k$ $$y_{k+4}-6 y_{k+3}+8 y_{k+2}+6 y_{k+1}-9 y_{k}=0$$

$x _ { k + 1 } = A x _ { k }$

Calculus 3

Chapter 4

Vector Spaces

Section 8

Applications to Difference Equations

Vectors

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in this example, we have 1/4 order difference equation that's provided here, and our goal is to do the following. So our goal is to write this particular equation in the form of a vector. X K plus one equals a particular matrix, a times except K. So if we can determine what the Matrix A should be and what the vector except cave must be, then our work is complete. Let's do some early analysis on this. This particular equation is a Order four difference equation. That means our vector X K is going to have four elements all together. Next, let's use a theorem that's also going to guide our work. So the theory tells us if we have a difference equation of order N, then we'll have a vector x k that's formed in this way with four entries altogether, we start with y que then increase on the sub scripts on K. So the second entry is why K plus one. Then why K plus two? Finally, why K plus three. Now we're ready to describe the Matrix, and if we have a order, N and N is equal to four r matrix is going to be of size four by four. Let's make it appropriate size for that task. And what the rose air telling us in this theorem is will start with a zero will produce a one and then the remaining entries are all zeros. So I have 0100 as the first row of this matrix. Now, the theory tells us that the only difference amongst the Rose is that if there was a pivot here, the prove, it moves down to the next column and then we'll have zeros thereafter. So back to a matrix say will go 00 place or pivot here and places zero next. Next, go 000 and place the pivot here. And that's our 1st 3 rows. Now the next key part are these quantities that we see here in the final row. Notice the sub scripting the sub scripting tells us that will have in our case and a one to be negative. Nine. A two is six. A three is eight and a four is equal to six. So oh dear, rather than the sub scripting before our A one is negative. Six. A two is eight 83 6 and a four is negative nine. So don't make that mistake. The sub scripting increases from left to right, starting with E K plus three for the Order four Difference Equation. Then we go in the reverse order, starting with a four. Take its opposite. So we need the opposite of this quantity, which makes it a positive nine produced here. Then the very next one. We take the opposite of this value, so we'll have a negative six. Likewise, take the negative off A two for a negative eight. The negative off a one gives us a positive six. And now we have produced the required matrix. So with this matrix given here with XK provided here, we now have this particular equation which captures the same information as the original difference equation.

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