write-the-difference-equations-in-exercises-29-and-30-as-first-order-systems-mathbfx_k1a-mathbfx_k-2

Question

Write the difference equations in Exercises 29 and 30 as first-order systems, $\mathbf{x}_{k+1}=A \mathbf{x}_{k},$ for all $k$ 
$$
y_{k+3}-\frac{3}{4} y_{k+2}+\frac{1}{16} y_{k}=0
$$

Nikolaos Alexandrakis · Accepted Answer

we will right this third or their difference occasion as the first or the system in the form X killed plus one. We will write it with over line just to know that this is a vector. It was a times ex gay Which way is three is three times three mavericks Since we&#x27;ll have the thunder, their system and we proceed by considering the vector x t Uh, it&#x27;s consistent these three values and the vector x k plus one which is this vector for following the definition off ex gay and we write Get with the foot taking the talk on the information from thing this and get some of them got played Tow the federal court in areas see off Gabe Plus three equals three over four CK plus two minus one over 16 ck and we perceive as in the book. And, uh, that&#x27;s an example of the book. And we write it as they seduce them in the for month eight times ck where a equals one or one 01 and minus one over. So extreme. Oh, and city over four. All of these times, the vector X Games

Michael Jacobsen · Answer

in this example, we have 1/4 order difference equation that&#x27;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&#x27;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&#x27;s use a theorem that&#x27;s also going to guide our work. So the theory tells us if we have a difference equation of order N, then we&#x27;ll have a vector x k that&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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.

Charles Carter · Answer

That&#x27;s the function y equals negative 25 over three acts and just wants it written in the form y equals K over axe. Um, so the best way to do this is just a ship that three up to the numerator S o r. Final answer here would just be my equals. Negative 25 over three. And ah, just can&#x27;t stick that in parentheses, extra money that that whole expression there is a k and then over X. So I can just help you with transformations and getting the graph of that correct case of y equals. Ah, no. 25 over three over axe. So you also think about like a is just negative. 25 over three. Okay. Thank you.

Alexandra Embry · Answer

in this problem. We have a system of three linear equations, and we want to write it in the form A X equals B, where a X and B euro matrices and RB matrix is going to be army tricks of solutions of our constant solutions. And our X matrix will be a matrix of variables. So that goes ahead and starts this out for us or variables are X why and Z So here&#x27;s our X matrix and RB matrix is going to be what&#x27;s on the other side of the equal sign what we want our linear equations to be set equal to so negative three native to and negative six. All right, so to multiply, um, some matrix A by a one by three matrix and get out of one by three matrix. We need this to be a three by three matrix. So we&#x27;re gonna have nine entries in here, and we&#x27;re gonna just do this row by row, so you&#x27;ll notice that when you multiply matrices So when we multiply this a matrix by this ex matrix will multiply this row by this Carl, Um and we&#x27;ll get out our first solution. So we want our first solution to be equal to negative three slut means we&#x27;re looking at this first linear equation up here and we want to be multiplying one by X three by why and four busy. So that exactly gives us what should go in this first row? It should be 13 before, and then we&#x27;re going to do the same thing. We&#x27;re gonna dio that this linear equation is gonna go here because we&#x27;ll multiply this row by this column and get out this entry right here. And we want it to be hold a negative to. So we just put in the coefficients to this linear equation. We want one x two y and three z. And then for the last one, we&#x27;re gonna do the same thing. So one x for why and three z So we can double check the press, the right thing. We get out. So let&#x27;s multiply these This a matrix by this ex matrix. So we get out one X plus three Why plus four c is our first century. Our second entries this road times This column so X plus two y plus three z in our last century is our last room well supplied by the column X plus two y plus three z We&#x27;re sorry X plus four way. So this should be a for okay and thes Um, we want this to be equal. Teoh. Negative three negative two and negative six. So this gives us exactly are linear equations up here. So we want breast first equation to be equal to negative three are exactly what we have. Second equation equal to negative two In our last equation equal to negative sex, it&#x27;s exactly are linear equations. So our final solution is that a is going to be 134 123 one for three. Our X is X y Z and R B is negative three negative to negative six.

Charles Carter · Answer

this room gives us equation three X y equals 12 and it wants us to read it in the form y equals K over X. So in order to solve this and first woman divide by three. So just have X y equals spore. Went about him by Axl have y equals for over X stark. A value in this case is for thank you very much.

Nikolaos Alexandrakis · Answer

we need to solve that. This special signal satisfies this situation and then find that enough solution off this defense Security to begin with, we just substitute this white gate of vacation and welcome to you. Bless to my nose. Class three off the door. D&#x27;oh! Thanks. Gay plus one minus four. My nose toe. Okay, my nose. Four. And after standard calculations, we arrived to the conclusion that these evaluates, uh, one class three K, which means that this is indeed the special solution sophistication. Now, in order to proceed the way, need toe, find the final solution of the Hamas EMS occasion. Which means that if we forget that Uncle machine a sterile and the substance of places with zero what solution? Kind of way. What solutions? Down with flying. We proceeded by the theory presented in the book where we construct the oxygen education with We assume that the solution because the form, like a good start for gay and because this form then gave us that&#x27;s for execution, uh, was oh, other than zero. So we need to find solutions toe the second vacation. Now proceeding by standard high school theory, we find that the second of dedication has this solutions. One door, which evaluate oneness positive and vibrates at one over two and the other one is negative and evaluates up minus two. So they&#x27;re not homogeneous like you&#x27;re some he solved by this general. Thanks soon with C one and C, it&#x27;ll being arbitrary. They are numbers plus superb minus two. So they k no, they&#x27;re not homogeneous. One has a general solution off this form. The home a Gino&#x27;s plus a special solution off the non coziness Jason we seize, which produces this formula for this occasion. And now, in order to proceed, we just need toe find to find out that these two solutions mind stood three K and one overto 18 at linearly Independent. And this is standard analysis. We&#x27;ll just evaluate the concert. My tricks with this test can be found on Page 2 47 It&#x27;s known as the personality Test, and it basically evaluates the solution. Started schooling off the mathematics called like a certainty. Methinks and its column is this. The solution is the solution that Gillian Kaye plus one and calculating the government off the smart tricks. We find out that this is not zero so we have linearly with enough independence off this two solutions off this. So the solution to this problem a quince at C 11 over to to the give Listen to mine stood three K plus the special solution which is this one for C one and C to arbitrary serial numbers.

Problem

Is the following difference equation of order 3$?…

Problem 30 Medium Difficulty

Write the difference equations in Exercises 29 and 30 as first-order systems, $\mathbf{x}_{k+1}=A \mathbf{x}_{k},$ for all $k$
$$
y_{k+3}-\frac{3}{4} y_{k+2}+\frac{1}{16} y_{k}=0
$$

Answer

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

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Heather Z.

Michael J.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Anna Marie V.

Heather Z.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications