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14. For thc following linear diffcrential system, determine conditions on the parameler a such thal the origin is a (a) saddle; (b) stable nodc,or (c) stablc spiral...

Question

14. For thc following linear diffcrential system, determine conditions on the parameler a such thal the origin is a (a) saddle; (b) stable nodc,or (c) stablc spiraldx dlt dy dt(a _ 1)x + Y,~ay

14. For thc following linear diffcrential system, determine conditions on the parameler a such thal the origin is a (a) saddle; (b) stable nodc,or (c) stablc spiral dx dlt dy dt (a _ 1)x + Y, ~ay



Answers

Given the system of differential equations $d \mathbf{x} / d t=A \mathbf{x}$ , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral?
$A=\left[ \begin{array}{ll}{-1} & {2} \\ {-3} & {0}\end{array}\right]$

Okay, so let's start looking at the new all kinds of this problem. I'm going to do the external client in blue head wounds. Excellent. Klein in blue. So for the external Klein, we have the equation. Negative three x plus y is equal to zero. Okay, so that's the first equation here. Right? So we have Why is equal to three X? Hey, so that's a line going through the or dream of the slope of three is gonna look something like that. Now for the second equation or for the wine? No. Klein. Right. That's gonna be the equation to X minus two. Why is equal to zero or just why is equal to X. Okay, so that's what I look like. This here. Okay, so now let's plug in a test point. Let's say our test point is let's plug a test point here. So for example, 10 Okay, 10 We put that in. So negative three times one plus zero equals negative three. So then, um, let me let me put that in blue again. Zero is equal to native three. So that means we'll be going to the left at this point here Okay, so everything to the right of the blue line will be going left. Okay, Um, and then since it's linear and continuous, um, then that means everything to the left of the blue line is going to the right. Like so. Okay, now, um, now, let's do the red. So we have two times one plus, uh, zero again. This is gonna be too. So here at this point, everything's going up. So that's positive. Means going up going up, then on the other side, everything is going down. Okay, So combining those together into green lines well, here, So these are the directions our answers will be going in. Right? So if you can see here, uh, all of our answers are going or tending towards the origin, right? Mostly going towards the origin. So we take a few lines phase lines, for example. Um, we'll get you know, either just going straight to the origin or maybe going around like this, you know, some just going straight to the origin. So our baseball it looks something like this. So this is going to be a node

All right, so let's start by graft. Mental finds. So for the excellent Fine. Here we have just the equation. Why? Equal zero? So that's gonna be a horizontal line here and then for the wild Klein. That's gonna be this equation here. So this is the lion X equals zero, which is a vertical line here. So let's plug in a test point over here. Let's say 11 Right. So if we play in 11 so zero times one plus one times one £1.1 we get one which is greater than zero. So above the blue line, everything will be going to the right. Thanks. We have going to the right, going to the right, um, there and then below the blue line. We have going to the left now for the why. So plug in 11 Here. One puns, one plus zeroed happens. One. This is gonna be one being greater than zero. So here will be going up. Oh, also, huh? Then here, to the left of that will be going down, down, down, down. So let's combine all those together. We combine those, we kind of will get this, um, perv here. Okay, Hopes. Here we go. So this is going to be a saddle

Okay, so we'll start this problem by graphing the no clients. So first, let's grab the Exhale, Hein. So the first equation cause as X plus Y is equal to zero. So we get that y equals negative X. So the exile fine is gonna look like this here. Now for the white knight plane. Right? That's gonna be the second line here. So that's gonna be zero X plus Y equals zero. So that's just gonna be the horizontal line here. Mmm. Now, let's put in a test point. I'm gonna pick 01 So you're one here, so if you plug in 01 one times zero plus one times, one is equal to one which is greater than zero. So that means all the points above the blue line will be going to the right. Okay, then see the left below the blue line. Well, now let's plug it into the second equation. The 0.1 So zero times zero plus one times one is also equal to one which is greater than zero. So here we're going up. Hey, this is all going up and then below the red line, we'll be going down. So it's combined. Those here. Okay, So if you take a look at solutions here, they seem to be all going away from kind of going away from not this one here, away from the note, but kind of any somewhat linear fashion. Um okay, so this one's gonna be just a note. Well,

Okay, so here we need to concern the vase plane with annul Klein's and equilibrium. All right, so we're are looking at the region X y greater than equal zero. But first, let's do the, um, the excellent client. So the excellent clients are when ex prime is equal to zero, right? Able to zero we saw. For this, we have two different equations. Will have X equals zero and three minus x minus Y equals zero. Or why is equal to three minus X not for the y equilibrium. We're going to have two different known clients as well. Y equals zero and two minus X minus Y equals zero. So let's graph the region X greater than or equal to zero. Okay, so let's start with our expel clients. We have X equals zero. That's when this flying here and then three minus X is going to be, um, so I'll say three is up here. Okay. Now for the wine, all clients, Why call zero is gonna be down here. Okay. And then to minus X is going to be, uh, this line here. Now, the equilibrium points are worthy external client and the wine elk lines intersect. So they're gonna intersect Here, here and then here. Those were the points. So for part B, those are the points 00 02 and then 30 Now for the directions. Emotion. Now we need to, um, plug in some test points to get the direction of motion. Um, So, for example, if we plug in Okay, the point here, this is 11 for example. So if we plug in 11 into both of our equations, so let's do it for the ex first. So one times three minus one minus one, for example, this is going to be one times one, which is one which is greater than zero. So in this region, we're going to be increasing. Uh, here. So this is always gonna be increasing. Increasing, increasing. And then now, uh for why One times two minus one minus one. Oh. Huh. Um, actually, this is going to be instead of 11 I'm gonna actually choose the point 1/2. 1/2 my bad. This should be the point 1/2. 1/2. So 1/2 times three minus 1/2 minus 1/2. This is going to be equal to one, So it was still greater than zero. Okay, now, if we do 1/2 times two minus 1/2 minus 1/2 this is going to be 1/2. Uh, that should be actually read No 1/2 times two minus 1/2 minus one. How it's going to be equal to 1/2 again. Still greater than zero. So we're gonna be increasing in this region? No, for the outside. Uh, well, we can choose here. This is going to be, let's say, 1.251 point 25 maybe, Um, if we choose our next another point in here, Let's see. Um, or actually, we don't really need to. We can just do a point outside here. Let's do, um to to. For example, 22 should be out here. Okay. Three minus two, minus two. Um, Okay, so 22 should be out there and then Okay, so let's plug in the plane to two. So we have two times three minus two, minus two. This gives us so minus three. Minus four is negative one. So we get negative too, which is less than zero. So, in fact, we are going to be decreasing this way and then for okay to minus two. Here we have negative four times negative, too. Um And then So we have negative two times two, which is going to be negative for again. Less than zero. So we're going down, Down, down, down, down, down, down Combining all of those right? We have this way, this way, this way, this way. So, um, I'm gonna go ahead and remove that one. So these are the directions emotion for all of our regions, and we're done.


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