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Solve the following dynamical systems_ Find and classify any equilibria _ Un+1 (2/3 Un; @o = 6 0n+-1 (4/5) an + 100, 20 Un+l (-6/5) an + 500, @o 25 Un+l 2 a, 30 , a...

Question

Solve the following dynamical systems_ Find and classify any equilibria _ Un+1 (2/3 Un; @o = 6 0n+-1 (4/5) an + 100, 20 Un+l (-6/5) an + 500, @o 25 Un+l 2 a, 30 , ao

Solve the following dynamical systems_ Find and classify any equilibria _ Un+1 (2/3 Un; @o = 6 0n+-1 (4/5) an + 100, 20 Un+l (-6/5) an + 500, @o 25 Un+l 2 a, 30 , ao



Answers

$$\begin{array}{l}{24-25 \text { Find all equilibria and determine their stability proper- }} \\ {\text { ties. Your answer might be a function of the constant } a \text { . }}\end{array}$$
$$x^{\prime}=-x y+y+a x, \quad y^{\prime}=2 y-x y$$

Be given the matrix A and defector deep. So when you saw for the general solution of this, you should get this is your answer for the general solution. I never really used method of variation of parameters. So we're gonna solve the system each one of people each to teach each to the A P uh, Times X would be so each one each to each. When you solve this system yet see one prime prime and CP problems in integrating at 1 23 and our particular solutions just the one time, each one of the few times each people three you can think of that. It's just your general swishing exit P Times defector, with all of your with all of your conference, the one you see three. When you do that multiplication we should get This is the general your particular solution. And then to get the final mission, you just add the general solution Pft

So here we are asked to write different constants. So and there according to chemical equations. So we first have HD two plus reacting with for ammonia, leading to the formation of a complex I in so this is just like any other equilibrium constant concentration of products or reactions accounting for stock geometry. In our second we have a 10 complex sign, So we have 10 to 4 plus reacting with six fluoride, still leading to the formation of snF six to minus. And our formation are instantaneous constant. Would be the concentration or complex on the product. However, the concentration we react, it's and we're repeating this for our third complex on which is after he three plus plus, sexy and minus, leading to the formation of a complex sign. So are instantaneous contact, instantaneous constant. And this gives our final answers.

Okay, so I have these three equations, and I noticed that if I just combined, I'm looking at the coefficients. Other in here I can add or subtract automatically. Cancel variable. Well, I can add one and three together and that'll I met the cancel Z 10 x plus. There's no ax here, so it's like it's there's a zero. X is 10 x plus eight. Why disease cancel therapist to will be to. And then I could divide each term by two. Since they all have a factor of two, which would be five x plus four y equals to develop it too, is one. I also notice, um, I'm staying on the cancelling Z theme. I could multiply the first equation by two to get six Z in order to cancel with the second equation. And I need to work with the second equation because I always were already worked with the 1st 3rd and I need toe change of what I'm doing. So two times the first ones is 20 x plus four y minus six z equals two times zero is zero plus the second equation, um five X plus four y plus six z equals negative one and adding these together. So 25 X plus eight. Why disease cancel equals negative one. Now, I know I want to cancel either an X or a y, so I'm gonna multiply it. I'm gonna cancel. Why? I'm gonna multiply this by negative to negative 10 X minus eight. Why equals native one when I combined these wise cancel get 15 x people's negative too. Um, excuse me. This is negative two. So then this will be actually negative three. And I'm going to divide by 15 to both sides. Um, and this will become X equals ***. 3/15. Fight about. Divide by three to the numerator and denominator. That's negative One. So it X's negative. 1/5. I'm gonna plug this back into the green equation to solve for y. So five times negative 1/5 plus for why equals one five times negative. One fit the fives cancels. That will just be negative. One plus for why equals one at one to both sides for why equals two divided by four. Why will be one half Okay, So I have X is negative. 1/5. Why is one half and I want to solve for Z. Um, I know I'm just gonna plug it into the first equation, but I could plug it into any equation. 10 times negative. 1/5 plus two times, one half. Plus it'll excuse me. Negative three z's native three times e equals zero, 10 times negative. 1/5. That's the same as multiplying the new writers and denominators of two fractions. So this will be negative. 10/5, which will be negative too. Plus one minus three Z equals zero. I'm actually gonna add three Zito both sides to get Z by itself and simple find the right side native to plus one will be negative +13 z, some to divide by three z will be negative one third. So therefore, I am going to have one solution that crosses at three points negative 1/5 1 half and negative one third

Okay, so I have three equations here and my goals to solve for one solution or symptom. Infinitely many solutions. Nesa Lucien. Um so my first goal is to try to get one variable by itself. Since my 2nd and 3rd equation have, um, only two variables, I'm gonna solve them like a two way system. So I'm just gonna add those two equations together. When I do that, when it add these two together, I'm gonna end up getting to why disease cancels equals two the bad by t to both sides. And then why will equal one? So I know that why equals one? I could plug this back in Teoh either equation two or three to salt for zeal. Plug back into two third place. It's y plus Z equals two equals to replace. Why, with one plus c equals two struck one. Besides, E is one. I have wise ones es one. And now I'm gonna play these both into the first equation To solve or so x plus y plus Z equals five x plus one plus one equals five and then I'll end up with X equals three. So therefore I have one solution and it's +31 What


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