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Consider the 2-dimensional system of linear equations~=l- 'Jx Note that the coellicient matrix for this System contains prameter Determine the eigenvalues of t...

Question

Consider the 2-dimensional system of linear equations~=l- 'Jx Note that the coellicient matrix for this System contains prameter Determine the eigenvalues of the system in terms of The qualitative belnvior of the solutions depenks tlie vul of It changes fromn ollc ypve aiot hiet (olso whcn but (his poblem lccuses On the change 4 0). Classily tho euilibium point o the sstem (by type nnd stability) Wn 0.wlen Wa Determine the cigenvectors lor the system (in ters of &) for the cnses -4 <

Consider the 2-dimensional system of linear equations ~=l- 'Jx Note that the coellicient matrix for this System contains prameter Determine the eigenvalues of the system in terms of The qualitative belnvior of the solutions depenks tlie vul of It changes fromn ollc ypve aiot hiet (olso whcn but (his poblem lccuses On the change 4 0). Classily tho euilibium point o the sstem (by type nnd stability) Wn 0.wlen Wa Determine the cigenvectors lor the system (in ters of &) for the cnses -4 < 0 < and Determine tle nullclines for the system (in terms of a) tlie CAS whert Using information from the eigenvalues anch eigenvectors draw scquence of phase portraits for the systemt; for the follwoing situations; <0, 4 20 4 >



Answers

Consider the system $\mathbf{x}^{\prime}(t)=\mathbf{A} \mathbf{x}(t), t \geq 0,$ with
$$ \mathbf{A}=\left[ \begin{array}{rr}{-2} & {1} \\ {1} & {-2}\end{array}\right] $$
(a) Show that the matrix $A$ has eigenvalues $r_{1}=-1$ and $r_{2}=-3$ with corresponding eigenvectors $\mathbf{u}_{1}=\operatorname{col}(1,1)$ and $\mathbf{u}_{2}=\operatorname{col}(1,-1)$
(b) Sketch the trajectory of the solution having initial vector $\mathbf{x}(0)=\mathbf{u}_{1} .$
(c) Sketch the trajectory of the solution having initial
vector $\mathbf{x}(0)=-\mathbf{u}_{2}$
(d) Sketch the trajectory of the solution having initial $\quad$ vector $\mathbf{x}(0)=\mathbf{u}_{1}-\mathbf{u}_{2}$

In this video will go through the answer to question number 17 from chapter 9.5. So we're given a matrix. A Onda were told that, uh, told consider the system Ex prime is equal to a thanks. So it's two by two matrix on We asked Thio, do a series of, uh, turns a series of questions about this system. Simply a rusted verify the Eiken values. And I can vectors so we don't actually to drive them. We just need to verify that they are indeed at the Eiken vectors and I can blow you off this matrix. So all we need to do is show that a minus one times I That's that times the identity matrix Times by the fist. I'm vector zero. Thanks. Let's do that. So a minus R one, which is to have that and the matrix is minus one, 33 33 minus three times by you want, Which is the column factor, Ruth 31 This is gonna be the talk. It's gonna be minus three aside. Minus week three. Close Route three, which is zero on DDE 33 times 33 which is three minus three, which is ever so At least I wanted you. One other are and I can vector. And I could buy you night and I connected. Let's try to show the same for are too on you too. Artoo is minus two. So one minus minus two is three. They're got ring three 33 minus one plus two. This one on DDE. We're told that the Eiken vector is one minus 33 Okay, so three minus 33 sides withdrew. Three, which is 33 months to me is ever on every three months. 33 zero. So yeah, two on dhe you to also, and I value and I respect it. It's too muddy matrix. So these could be the only I'd buy youse, like, practice off the matrix. OK, Poppy, this is gonna be helpful. Now, before we're get ahead with the sketching off the trajectories, it could be helpful to write down the general solution off this equation, which we do because we have, uh, a linear system linearly in many way independent system of high Quebec list and the corresponding back in values. So it's gonna be a constant C one times by E to the first I am value, which is to times to you times by jag Vector, which is big three one who was a second constant times e to the second I compare value Remind us too. T times a second, Aiken Vector one minus for you. See? Okay. So I have a mind to sketch the trajectory solution with initially a vector reminds you want. Okay, this grid. So what is minus? You won't look like so you want. Is this guy so minus that is gonna be minus root three rue three's between one and two. What would it be? About 1.7 ish. Um, so minus 33 minus one is gonna be somewhere like here. That's gonna be where, where? Initially starting. And since we're starting with a particular item factor far from multiple events just to scare the multiple of annoying factor, that means that weaken only move along as time goes on, we can only move along that I connect it. So for the minute we can forget about this guy. And as time goes on, this is just gonna get bigger and bigger and bigger. So as time goes on, we're going to go more and more in the direction off. AA minus. You won't. And the doctor minus the one is this way. So we're just gonna keep going. Keep going. Keep good. Bond is gonna keep going in that direction. Faster and faster rate as time goes on. No hotsy. What about when we stopped with initial value of you too? So you two is the vector one minus route three. So one minus 33 is here somewhere again, looking at this general solution. As time goes on, we can only stay in the direction off. You, too. On. As time goes on, we're gonna slowly get closer and closer to the origin because e to the minus two tost gets bigger, Bigger, bigger, closer, closer to zero. So that means we're gonna start here and we're gonna get closer and closer to the origin. As time goes on, we'll never reach it because he's the power of Manitou. T will never actually hit zero. We'll just get closer. Closer, Closer. Okay. What about pot? De So says we start, Huh? You two minus you want So you two remember Woz here on Dhe minus. You won't is over here somewhere. So you two minus you won't. It's gonna take us to Well, if this is you too, and this is a minus. You want then up those two together and you gotta get over here somewhere. So that's our That's why you two minus you want. Okay, so it's time for his own. We're gonna have more and more of this guy that's gonna get bigger, bigger, bigger. And we're gonna have less unless of this guy. So we're gonna start off with a bit of both of them, so we're gonna start off going kind of more in the direction of both of them. It's kind of this way away from dark day away, regularly away from the origin. But then the component off you, too. It's gonna be much, much, much less than you want. So it's going to start curving into the direction off. You want something like that direction

In this video, we're gonna go through the answers. Question number 35 from chapter 9.5. So, in part, A were asked to find Ah, what's to show that the matrix A here has repeated I value our equals two minds Well, on dhe show that all the time it is. That's the form given in the question. Okay, so I start with the wagon values. So we need Thio. Find the determinant off the matrix A minus. Aw, times dead and see matrix. That's this. Determine in here. That's gonna be well minus ah times by minus three minus. Uh, because for that's gonna be able to u R squared. Ah, hoofs every, uh, minus. Ah, sets plus thio minus three plus four plus one. Okay, so this is easily fact arised as ah course one squared. If that's equal to zero, then we have repeated Eigen value eyes equal to minus one. Okay, cool. So now it's fine. The Eiken vector associating with that. So a plus the identity matrix is gonna be too two minus one for minus three plus one is minus two okay, times by infected, you go on is equal to zero So therefore, any item vector must be proportional to Okay, let's see if we let the first component be one second component. Must be two cases going breaks that fulfills. Ah, well, the quiet for a movements part B. Uh, so this is quite easy. So the an intravenous solution can be Rin as eat first I about you East the first time value, which was minus one times t times by dragon Vector 12 And it's very easy to check with us. Solution off the system, given you a question. Okay, But see, so I guess a little bit more tricky here for us to find a second mini independent solution. Eso we have only we had a repeat. I can value said can't just use the second night value because the repeated I could buy you only had one linearly independent. Uh, Ivan Vector. So it's Astra's use. The form next to is equal to t eat. The minus t does buy you one plus even my honesty times by you too. Okay, so we're gonna substitute that in. We're gonna need to find the first derivative using the what a fool on the first term is gonna be one minus t times e to the minus t Because it's the modesty. Different shades to minus eats. The modesty does buy you want because it's just a constant vector. Minus it reminds t you too, it Okay, so, saying, Ah, you two prime people to eight times you, too. So meaty. Prime waas one minus t Because by e to the minus t you want minus eight minus t you too. And then a times You too. Uh, well, that's just gonna be tee times e to the minus t times by eight times by Yeah, yes, I was by a you one. So say you one because you want is an aiken defector with Aiken value minus one and a you want he's gonna be equal to B minus one times You want. Just by the definition off, you won't be Knight director of the Matrix egg. Uh, we got close E to the minus t times bite matrix. A YouTube look. Okay, so let's have a look. What's going on here then? So this minus t eat my honesty? You want concussive with this modesty here? And then what we're left with is ah, eats the minus t You want minus you too equals eats the minus t a You too. So eat the virus Taken Never be zeros. We can cancel that. So this is gonna allow us to arrive at a plus. The identity matrix I times by e t equals you are. Okay, so now we console that. Let's have a look. At what a plus. I Yes, that's gonna be the matrix to minus one for minus two times by you, too. He calls when you want. Was the vector one too. So this means that you too. Okay. So if the first component is one, then we're gonna have second opponent is gonna be two times one minus one, which is just what? So that's, uh, a solution for you, too. Okay. Now, huh? Day as us girls just find what a close I Yeah, like squared times. The YouTube is okay. So this is just a plus I times by a close eye. That's just the definition of squaring. Something just most quiet by. You must buy twice by that by that thing. Okay, So then hey, they were just showing that April's I times you two is You want. It's that in this Caribbean is a close eye you want. But it was high time too. You want because that you wanted a infective. Ah, metrics. I with Ivan value minus one, that is just gonna be equal to zero. And that completes the

In this video. We're gonna go through the answer to that question 39 from chapter 9.5. Um, so yeah, but given the Matrix A which is written here, we asked Oh! Ah, derive. Uh, yeah, There is an ancestor questions A, B, C and D. I think this is quite hard questions, so I'll try and take us through it slowly. Okay? So question apart a were asked to find that I can values I come back to us basically on dso first. Find ion values. Gotta find the determinant off the matrix A minus I So basically mind seeing are from the leading diagonal. Okay, So what's that gonna be? We can evaluate it around the top itself forever. That's two minus r times by the matrix of at the seven of the Matrix to minus one minus two minus one minus. One times the matrix. December of the Matrix. No, there should be a minus. Other close. One times the matrix. 12 months, huh? Minus two minus two. Okay. Uh, I think I'll avoid boring you with the algebra for this one. Um, you guys can work that through and show that is equal Thio. Well, you can expand it all out and show that it's minus, uh, cubed. Plus three squared minus three. Ah, close one which could be fact arise to be one minus R cubed Get. So that's equal to zero than that shows the the value of ah, which are Ah, I convey values. Eyes equal to one on It's a triple. Really? Because about three here. So it's an egg value with multiplicity three. Okay, fine. Victor, that's fine. I'm vectors. We find a minus one times. I because I value this one times by you equals zero. So what is a minus? I was going to the back there. The matrix one 11 one on one minus two, minus two minus two. It comes by u equals zero. Okay, so if we let the components off, you be X y zed, then each of these rows in the matrix equation tells us the experts Why? Course said is equal to zero. Okay, so then what could solutions look like to this equation? Well, we could have that. The first term at the Ex Capone is zero. Then that would mean that Ah, the wines that components with a sign of each other. We could have that The y component is zero. In which case it'd be minus one one. Or we could have a second potent be zero. Gonna be zero minus one, but one. Okay, but now look at this guy. This equation this factor could be written in terms off this factor on this vector. So you can see that if you, uh let's see if you do this vector minus this factor that will give you this vector. So therefore, this factor is not linearly independent to these guys. So, um, where these guys are linearly independent to each other. So therefore, any item vector must pay off the form. That's, um, constant s times by the first of the linearly independent vectors. First, some constant v times a second of our linearly independent backers. Okay, so if so, have chosen this kind of arbitrarily because we could equally right. We could have equally written this guy as a linear, some off this thing guy and this guy, in which case we would change the inspector's on DDE. That would also be correct. So we've kind of written this kind of arbitrarily the point is that we kind of have only two degrees of freedom on our choice. If I came back that way, can't write the Eiken vector as a some off Constance Well supplied by all three of these. I mean, we could, but it would kind of be, ah, were necessary because we already showed that, uh, to sufficient. Just two of the of the expression, too. Give you all of the possible high in vectors to the Matrix. A. Okay, so hopefully that's Colbert. Why? We just have two vectors here. Okay, So, uh, B, this is just following standard rules for linear started different differential equations from those two aiken vectors. We can write too linearly independent solutions on They're gonna be e to the First Aiken, How you bought the only item value times T c to the t times by the first heart of the Eigen vectors on then. Secondly, Secondly, nearly defendant solution that's gonna be eats the tee times by the second darken vector. See? Okay, let's use the form that they've given us, which is x three, because t into the tea. You three plus a to the t you for. Okay, So what's the derivative of this Well, we can take out you take The city is a common factor. Used the product rule Gonna be one plus t you three plus you for times e to the t. We know that this is gonna be equal thio because what we've assumed that it's that this form is a solution to, uh, the matrix differential equation so we could have tea. That's a you three for us. Hey, you four times by E t t. Okay, so this guy and this guy are equal to each other so we can cancel each the teas and then compare coefficients off. Tease will tell us that a U three is equal to you three itself on and you have a cross compare coefficients here. This tells us that a minus by you three is equal to zero on dde a minus. I you four secret to you three. Okay, so this tells us that you three is an icon vector. That means that it needs to be of the form. Found a pot A which is s times minus one 10 close a times minus ones. They're well you asked me about for so Okay, let's try and figure out what form we're gonna take. Okay? So to figure out what you four is gonna be a we're gonna need Thio. Find out what this matrix is. A minus. Eyes the matrix one 11111 minus two minus two minus two. What time did not buy you for? And then this is gonna be equal to you three. Which, as of yet, we're unsure as to what I infected to choose. So what's gonna help us? Well, if you four has the components X y and said Just try to see you. What's that? You for its components, X. Why? And said then the top road off this equation, it's gonna tell us the expose, what was said is equal to whatever this first component is a man. The second I was gonna tell us the extras Wipers said because he was the second component third component in the third row is gonna tell us that minus two times by extra swipe, all set is equal to Okay, So then now, in order to make this house a any solutions, all we need Well, we need this component on this components were the same because otherwise would have experts watches that is equal to two things which is just mathematically consistent. So that means that, um yeah, I need those guys to be the same. So let's say that we let them be equal to one on one. Let's see whether weaken do that. Uh, yeah, definitely. Can. So that some if s is equal to you. Uh, let's see, that's equal to one. And B is equal to minus two. Then that will work. And then how does that work? That tells us that this 3rd 1 is gonna be minus two, which shows that that's gonna be minus two 11 works because now, diesel, these equations here or class into one equation which settles wipeout set is equal to walk. Forget it. So I've chosen you three to be 11 minus two. So therefore X, with y plus said where we need to choose. Yeah, a basically any backdoor that it satisfies this so we can just easily choose a vector 100 because that satisfies the equation. X plus y equals said Okay, so just finishing that off on these x three is equal to t It's the tea. 11 minus two. I saw x three. Plus it's the tea 100 we saw explore. Okay, then pot de a minus. I squared times you for equals. I was gonna be a minus. I times a minus I you for which is a that which is you three. And then we'd know from hot See, that is equal to zero. Okay, nothing. She's answer a question with the night.


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