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(2'; points) Find the cquilibria of the autonomous diffcrential cquations and evaluate the stability (stable unstable). Be surc show your #ork on how Yuu umive...

Question

(2'; points) Find the cquilibria of the autonomous diffcrential cquations and evaluate the stability (stable unstable). Be surc show your #ork on how Yuu umived Hour answcn (You only need of the mcthods: )a =1- ejxax = 1-3

(2'; points) Find the cquilibria of the autonomous diffcrential cquations and evaluate the stability (stable unstable). Be surc show your #ork on how Yuu umived Hour answcn (You only need of the mcthods: ) a =1- ejx ax = 1-3



Answers

$11-12$ Find the equilibria of the difference equation and classify them as stable or unstable. Use cobwebbing to find $\lim _{t \rightarrow \infty} x_{t}$ for the given initial values.
$x_{t+1}=\frac{4 x_{t}^{2}}{x_{t}^{2}+3}, \quad x_{0}=0.5, \quad x_{0}=2$

Here we have the differential equation. D Y de acts equals why Tim's Y squared minus one. We just want to find its equilibrium points and determine whether each one is stable or unstable. Or I remember, um, to find equilibrium Points, if you have a differential equation of this form of the right hand side is only in terms of why then we confined the equilibrium points by just, um, setting this equal to zero, right? Actually finding where, um, the derivative of y with respect to x zero. Um, so if you want to find an equilibrium points, we just had this equal to zero. This is a product. This will only happen when y zero or when y squared minus one is zero. This last one left one just gives us y equals zero. Over here, we can rearrange it so y squared equals one. So why is it little plus or minus one? So our equilibrium points are y equals 01 and minus one. Now, for each of these, we need to determine their stability. And if you look in the section what this is is we make a number line and check whether do I d x is positive or negative on each interval, Um, cut up by these equilibrium points and then we see if, um if the function is moving toward, um, these a glittering points on both sides and it will be stable, Otherwise it's unstable. So to determine stability, let's let's first make our number line. And we only have three points that we care about our people over in points. We just need to figure out if, um do I t x is gonna be positive or negative on each of these. So when we are less than minus one, then the white term is negative in the Y squared minus one term is going to be positive for ex view, plugging like minus two. You square, this four minus one is three. So to the left over here is always gonna be negative in between minus one and zero The Y part. It's still gonna be negative. But the Y squared minus one is gonna be negative. Also is positive between zero and one. Um, the Y term is gonna be positive, and westward minus one is going to be negative. So we get negative, and if we are bigger than one. Then why is going to be positive? And Western medicine is also positive. Okay, that's what this means is that as we like, move along the graph. Um, since this thing is like decreasing it, it, like, moves away in this direction. Since it's increasing here, it moves to the right as we go to the right and we have somewhere things over here. And so then determine stability. The stable points are those where the arrow's air both kind of pointing towards it so that the function is gonna move towards this point. And so what are number line tells us is that why equal zero is stable, right? We have the arrows both pointing toward zero. That is, it increases to the left of zero in decreases to the right of zero so that it's sort of approaching it. And then the other two, um, both minus one and plus one or the opposite, right? Both errors are moving in the other direction, so these are both unstable

So here we're trying to find the equilibrium points and determine the stability for the differential equation. Do I d X equals C to the Y minus 12 y minus three. Let's remember to find our equilibrium points. What we do is we set this equals zero and we want to find the y values that make do I d act equal to zero. And since this is a product, this will only happen when one of these is zero. So over here we got That'll happen when y equals three here. If we move the one over and then taken out your longer both sides, we'll get why on the left hand side, enough for a lot of one is zero. And so our equilibrium points are just a zero and three and now to determine the stability. Remember, What we do is is like in cook one. We make our number line and we indicate our equilibrium points and we just want to check in each of these intervals whether do idea access, positive or negative. So when why is increasing or decreasing? So if we are to the left of zero on this left most interval then we're a negative number. And even the why is gonna be less than one, since it's one over e to something. And so the left products in D I. D. X is going to be negative. And why minus three new pregnant negative number is also gonna be negative. So they're both negative, and we get positives when they're in between, um, zero in three feet of the wise would be bigger than one. So the left term is positive, but the right term why minus three is gonna be negative, b negative. And then to the right of three. You know, the line minus one is gonna be positive. And now y minus three is also gonna be positive. So we get this and then you can think about as we let go along the curve. Since its has its increasing here as a positive slope, the function is like increasing, um, towards this point. But since this is negative here, um, the expression can wants to move this way, since that's like a neck of slow. And then we put a right arrow here for the positive slope on the last interval, and then remember, we can we can use this to determine which is stable and unstable. The stable ones are the ones where the errors are pointing towards it. That's why equal zero and the unstable are when the arrows are pointing away and there we go.

So the differential equation is a wide crime is equal to five. Why? Times two e two The power off negative white negative life minus one. So we know that why Hat is an equilibrium if y crimes equal to zero at my hat. So if we set this equal to zero, if we set this equal to zero, we end up with a white hat, has to eat the power of negative lifetime. Just one is equal to zero. So we know for sure that I had is equal to zero or to eat about negative. Why hat is equal to one solving this out We end up with why two hat is equal to 0.69 Suite 12 jacket. Wie hat is locally stable. Equilibrium or not, we find the derivative off. This'd equation right here. So differentiating with respect to why gives us a G prime off y is equal to 10 e to the power of native Why times of one minus y minus fire. So plugging in Jeez, equal to zero. We get g prime off. Why had one is, um greater than zero which implies that why I had one is equal to zero is a locally unstable equilibrium, plugging in G prime off my hat to we end up with a number that is less than zero to negative number. So this implies that why two is a locally stable equilibrium.

Can number thirty four d y o ver t X equals actual alkali minus one. But five months. Why a singer? We fired you caliber in points by sick the wild boar. Jax was there. So either this party zero is his part. So when this part of zero Why AE is why was eat right? National off is one one months ones there or why come the five? So those are the two equilibrium point and we want to find this stability off those two points we're planting your hair is like two point seven substance so far these videos and he Ah So let your log off. Why is something like that? This party is one he one. So my ex e when? Why sorry this mourners and Eve his part. Ah, that's love Life's mourners and lawn. So this part is going to be laxatives, and this part is going to be a positive. So it's this whole party's funds too. And for eat to the five part when e ead speakers and when wise beaters and e smaller than finds his forties a Peter, this party's veer is and one so this party's product it's positive And this part is going to be positive, too. Because why smaller than five? So when wise inside this party's gonna be positive? What is the first lines? Like Direct? What? Why smokers and E. Dace Perch is positive thiss partisan activist. So this partisan actor are support is an active was lies Vickers and five. But this party's positive discord is magic. We connected her again and it goes a rose. It was negative means is decreasing Android away from that point, can't It was positive increasing do you crazy? And this point why it was e Teo era point our way so is unstable You quit off you caliber and point in this point See Arab points toward if toe, like was five is a stable equilibrium point thirty four.


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