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Math 228, Winter 2021 Name: Nonhomogeneous Systems with Repeated Functions Consider the system x' =2x+y + 6e' ,y' = 3x +4y.Rewrite the system in matr...

Question

Math 228, Winter 2021 Name: Nonhomogeneous Systems with Repeated Functions Consider the system x' =2x+y + 6e' ,y' = 3x +4y.Rewrite the system in matrix form:Solve the homogeneous system[x]-l:] where the coefficient matrix is as above_Examine the nonhomogeneous term from (a) and use t0 create guess for the particular solution. Just as with single nonhomogeneous equation_ is not enough. What else must be included? Why?(d) Solve for the coefficients in the guess (rom (c) by substitut

Math 228, Winter 2021 Name: Nonhomogeneous Systems with Repeated Functions Consider the system x' =2x+y + 6e' ,y' = 3x +4y. Rewrite the system in matrix form: Solve the homogeneous system [x]-l:] where the coefficient matrix is as above_ Examine the nonhomogeneous term from (a) and use t0 create guess for the particular solution. Just as with single nonhomogeneous equation_ is not enough. What else must be included? Why? (d) Solve for the coefficients in the guess (rom (c) by substituting it and its derivative into your equation in (a)_ (e) The general solution t0 the original system is the sum; []-[]-B]- Write it here.



Answers

Consider the linear systems
$$\left[\begin{array}{rrr}
1 & -2 & 3 \\
2 & 1 & 4 \\
1 & -7 & 5
\end{array}\right]\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]$$
and
$$\left[\begin{array}{rrr}
1 & -2 & 3 \\
2 & 1 & 4 \\
1 & -7 & 5
\end{array}\right]\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]=\left[\begin{array}{r}
2 \\
7 \\
-1
\end{array}\right]$$
(a) Find a general solution of the homogeneous system.
(b) Confirm that $x_{1}=1, x_{2}=1, x_{3}=1$ is a solution of the nonhomogeneous system.
(c) Use the results in parts (a) and (b) to find a general solution of the nonhomogeneous system.
(d) Check your result in part (c) by solving the nonhomogeneous system directly.

Did you know? Question number 45. Your first by water. And be one we need to intercept first and third. You're right. I want to change our tree. Yeah, My detained row at first place. They want to one minus form your second. Roy's further. These no defense Tral. But we want to make cereal at here. So probation is our to minus Phi. I am sorry. One and for the same tree. Forward probation. Ease our three births to water. Wine. Last Roy's Reduce your second Royce zero minus 16. Too late. Then determine Rois zero a minus one in minus pork. Right now. Interesting. Second and told. But you're right. Off who interchange our tree. More Fourth, Ruiz early dudes. All right, the tundra at second less. All right, the second right for this. Now we want to make zero at your sewer. Oh, patients will be our three minus sometimes. Sorry. Plus two times are too for your first and second rows are as it is. You're 16 minus 16 0 to minus 20 and eight mile estate cereal. Right now. Problem does a control. You can write immigration that he's eight minus story a dry. My inside is it will do for right. So you're eight wise equal dreams that minus four for your eyes equal to one by eight sec minus one by two. Now from this pastoral weekend, right, Your X plus two y no, I said is a core do minus four. No subject takes in this aggression. No, but the real you Why studies one by a that minus one by do to my eight that he's one my fault. And here to Monte family one by crudities one your negative for less one. There is a snag getting through and your negatives a minus one by for that that is negative by by floors that and there is no way I would NT eight The third column so we are said, is our free variable. So Tuesday perimeter T for said right So dinner solution snag 83 negative five by 40 Your eyes were won by 80 minus one by dude. And that is important D Thank you

Did you know? Question number 45. Your first by water. And be one we need to intercept first and third. You're right. I want to change our tree. Yeah, My detained row at first place. They want to one minus form your second. Roy's further. These no defense Tral. But we want to make cereal at here. So probation is our to minus Phi. I am sorry. One and for the same tree. Forward probation. Ease our three births to water. Wine. Last Roy's Reduce your second Royce zero minus 16. Too late. Then determine Rois zero a minus one in minus pork. Right now. Interesting. Second and told. But you're right. Off who interchange our tree. More Fourth, Ruiz early dudes. All right, the tundra at second less. All right, the second right for this. Now we want to make zero at your sewer. Oh, patients will be our three minus sometimes. Sorry. Plus two times are too for your first and second rows are as it is. You're 16 minus 16 0 to minus 20 and eight mile estate cereal. Right now. Problem does a control. You can write immigration that he's eight minus story a dry. My inside is it will do for right. So you're eight wise equal dreams that minus four for your eyes equal to one by eight sec minus one by two. Now from this pastoral weekend, right, Your X plus two y no, I said is a core do minus four. No subject takes in this aggression. No, but the real you Why studies one by a that minus one by do to my eight that he's one my fault. And here to Monte family one by crudities one your negative for less one. There is a snag getting through and your negatives a minus one by for that that is negative by by floors that and there is no way I would NT eight The third column so we are said, is our free variable. So Tuesday perimeter T for said right So dinner solution snag 83 negative five by 40 Your eyes were won by 80 minus one by dude. And that is important D Thank you

Hello there. So for this exercise we have these two systems of uh these two linear systems. But basically you can observe this response to the homogeneous system and this to the non homogeneous system. Yeah. So then the first part we need to find the general solution for the homogeneous system. So let's represent this as an extended matrix. So that means taking these matrix here. 000 Okay, but look, we have something happened here. The first the third bro is equal to minus the first row and the second row is two times the first row. So technically these two roads will eliminate by doing the following operations. Road operations. So if we take our two and then we say that our two minus two times R one and R three will be our three plus are one. By doing those operations. We obtain the following. So just let me copy this would write everything. So these two rows will become zeros 000000. So we need to find the solution for this system. You can see by the matrix that we have only one pilots. So that means that we have to free variables In this case X two and X three. So the solution will be dependent of X two and X three. So let's give some values to X two. Let's say that extra pickles 23 annex three. Uh is equals two. S. where T&S. Are real numbers. And having this in mind then we can find a solution for X one and X one is equal to one third times minus two. S. T plus S. Okay, so from this we obtained the general solution for the system and it's the vector that depends. Two variables. T. M. S given by one third. I'm going to take one third as a common fracture. And then we have here minus two T plus s minus two plus S three times T and three times as. So this is the solution. The general solution for the homogeneous system and actually we can write it better. And is that these solution? It's called X. H. For homogeneous surgeon. So X H t. S Can be written as the sum of two vectors. So will be T thirds time. Is the victor -230 Plus as over three as 3rd. 103 Great. So we have this and well we can replace this by some alter a constant. It doesn't matter. These are just as killer. So we can just replace right tms is the same solution. So this corresponds to the Alma genius solution of this system. Now we need to verify for the part B A particular solution. So we need to verify that the following vector 101 is a solution for the non homogeneous system. So that means to this system here so we copy this. Okay, so we need to verify that this vector here is solution of this. So let's check here. We're going to put the vector 101. So the multiplication of this will give us three, 6 -2 -3 Plus one. And you can observe that there is a result in 2: 4 -2. So this vector here XP is a particular solution of the system now. So for the part C we need to use these results, the particular solution and the homogeneous solution to give a general solution for the system. And that means that the general solution, let's say x g t s will be greeting us. The homogeneous solution plus the particular solution where the particular solution is the cost 101 and the homogeneous solution depends on two variables, T times -2, plus S one 03 So we just need to put the things together. So the general solution X G depends on T m s will be T times the vector -2, plus S. The victory 103 And the particular solution that is the victor 101 And this is the general solution of this uh in a homogeneous system. No, we need to verify that actually this is true. So to verify that we need to solve this system, the in a homogeneous system. So we need to find a solution of this. So to do that we need to consider the extended matrix and that means taking this matrix here two, four minus two. Okay, again, we're going to perform the same operations as in the homogeneous case. So we take our two will be Our 2 -2 times the first row and the third row will be just some of the earth, the 3rd row and the first road. And doing that, we obtain the following. So these are become zeros and these 20 and zero. Okay. So we think this from this expression we again we have to free variables X two and X three. And the general solution of this system is also dependent of T. N. S. And it's Graydon us T times -230 plus S 103 The particular solution in this case is given by two thirds zero. Um So we need to verify these. These these also for response, we can generate the particular solution. Consider in C. In B. Sorry. Uh We can consider so. XP can be greeting us some value of T. And s. Of these particulars. That is clear. Well it's not directly clear but we need to compare this so we need to generate the filming vector 101 should be equal to this part here to be some of these vectors. Yeah. So the first thing just pay inspection, you can observe the T. Should be equal to zero because we don't have any value here. Right? So T should be equal to zero. That's the first thing, this zero and we need a number one here. So that means that S should be equal to one third if S is equal to one third on the right hand side. We have the full we have the picture 1 3rd 01 plus the vector 2/300. And you can see that this is the battery 101 So there's a particular solution of deal in um a genius system.

This is a question of maturity. Seven. Your first by what entries would be once we need to divide room one by for So here on one do eight by Fuller. You know, for my foreign Want negative eight by forest. Negative to add my voice to and 24 by for in 86 second and third row Saar and ladies. Now they want to make cereal at here. So our oppression is our to minus two are one. And for the sentry, Prohibition is on three minus three. Everyone, your mysteries at the DEA's, you're secondaries. Judo minus two minus. Want anyone? And it's ordinary. 0 10 minus. They went in minus 29. Now we want to make zero and the sentry. So our operation is our three plus by times are one. So been here. First and second rows are ladies and detailed Release two years, zero, You know, getting prayer and negative before, right. So from the toddler, we can right here negative, very liquid to negative 24. And that gives That is equally do to right now from the second row here, we can write integrity to why negatives that is equal to one. So here we can write. Negative. So why is it will conduct bless one. Therefore Winik with negative big negative one. But he, uh that is according So you're right. It was negating to my do, right? No. From the first row, we can write your X minus two y plus two dead in the courtroom. Six So x is according to my minus through the last six. You're wise equal to negative three by two. And here's that Egypt will do too. So exit equal to negative three, never. 84 legs makes therefore exceeded quality minus one. They're controversial. Duchovny's mating warn Maybe you two by two. Thank you. Thank you.


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