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Consider the nonlinear system of differential equationsdxdy d =y-e"1y - TyDetermine all critical points of the system: For each critical point not on the y-axi...

Question

Consider the nonlinear system of differential equationsdxdy d =y-e"1y - TyDetermine all critical points of the system: For each critical point not on the y-axis: Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system Find the general solution of the linearised system using eigenvalues and eigenvectors. Sketch by hand the phase portrait of the linearised system_ Include all

Consider the nonlinear system of differential equations dx dy d =y-e" 1y - Ty Determine all critical points of the system: For each critical point not on the y-axis: Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system Find the general solution of the linearised system using eigenvalues and eigenvectors. Sketch by hand the phase portrait of the linearised system_ Include all working of any special cases corresponding to straight line orbits behaviour of general orbits as and t and slopes of orbits across the axes Identify the type and stability of the critical point of the linearised system Using PPLANE, produce global phase portrait of the non-linear system in a region that includes all of the critical points_ The phase portrait should illustrate the behaviour of orbits in the vicinity of each critical point.



Answers

obtain the solution of the following differential equations

In this question were given the differential equation Y squared plus two X y times dx minus x. Where do you buy? Is he going to 0? So we have to find an integrating factor for this and then solved find the general solution for this differential equation. So first I will expand this Y squared dx bus, two x y d x minus At Square Dy Physical zero. And we can write this as y squared dx and we can considered two X dx to be T x squared minus x squared dy equals zero two in the next step. And this stuff we can I understand that? We can notice that why times dx squared minus X squared dy is is the derivative of a caution. So yeah, for example the X squared over Y will be equal to denominator times the derivative of the numerator minus numerator times derivative of the denominator and divided by y squared. So if we consider why square to be the integrating factor, the wife equals one over White Square to be there. The integrating factor we could ride D X us. Why do you? X squared minus X squared dy divided by y squared is equal to zero. And then we could write D X plus. This is going to be equal to conservative and squared over by equals zero. And this means that the general solution of this differential equation is X plus X squared over. White is equal to the constant. So this is the general solution of this differential equation

The given differential equation is do you buy my dx plus two X. Y. Is equal to annex. We controlled by variable super blast Close, do you have a dx is equal to two X. You can take common five minus white. So it is 10 x minus two. Xy. So now separate the rebels. Do I buy five minus Y is equal to experience When you integrate. It's Ellen morph. I minus wife divided by minus one Because the coefficient of Y is minus an integration of two X S X square. With some constant of integration. So when you approximately by a line of five minus Y is equal to some constant k minus x square or five minus Why is equal to some kid ash heap ar minus x square. Or why is equal to five minus k dash par minus x squared. Is the solution Cash is an art to reconstruct. The same equation can be solved using first order linear differential equation. So basically this is a fast or a linear differential equations of the form divide by the explicitly affects into why is equal to cure affects. Now, if you compare your pdf, X should be whatever is multiplied with why that is two X and Q of X is 10 X. So what is the integrating factor? The integrating factories? E power integration of prof X D X. That is the power integration of two X dx is easy. Parks squares and integrating factor and the solution is given by Y times integrating factor. The integration of Q Times integrating factory eggs less. Some constant of integration. See so what will be by times integration factor Y times frx square is integration of a few times 10 X into integrating factory bx plus some constraints. See now this is pretty easy to integrate. Can use the substitution of X square is going to T. Then two X dx will be D. D. So fire can pull it out and two X. Dx exonerated as D. D. So it's basically Fipe R. D. D. Party D. D. Plus C. So what I get is fine to integration of your party is the party plus C. What is R D X squared? So it is five part X squared plus C. So what we've got now is wipe our X squared is equal to fi par X square plus C. Do I drove by the park square? So you get five plus C par minus X square. So it is the same solution we got. Second question also divided by D. X plus Y by X is equal to six X plus two. So you can start using a variable separable or uh like uh playing your differential equation variable separable is pretty easy again. So when you take and cnnpolitics divide plus wild dx. Is it going to six X plus two in two? X dx by cross multiplying this is actually it's a product rule. So it's D O X Y. So this is called A D O X Y. So X I'm treating it is available now and decide the 66 plus two in two X dx. Now we can just integrate both sides. What is integration of D of X. Y. It is simply X. Y. And you can integrate right side at six X squared plus two X plus some constancy. So X. Y should be six, execute divided by three plus X square, there's some content. So the answer will be two X cube plus X squared plus C. Now to use the linear differential equation mattered what is P of X here? X is basically one by X and Q. Of X is six X plus two. Q of X 86. X plus two. So let's find integrating factor. It's E power integration of one by X which is L N X and liberal next X. So it's Y into I mean the solution is given by Y into integrating factor. Is integration of Q. Types of integrating factor plus some constant of integration. And we end up with the same solution. X Y is equal to uh to execute plus X squared plus some concept of integration. See that's it. Third back the third party has given us something like this. So when you transport transport three to the left hand side we've I d x minus two X squared dx. Did you go to minus X. Dy? So what we do is X dy plus three Y dx is equal to to an x squared dx. Basically what you can do is multiply both sides with extra square. So this bit of extra square execute do I plus three Y X squared dx is going to two X power for dx. Now why I'm doing this because this is again a product rule of dysfunction D off executed by executing two D Y plus Y into dari video of excuses three X squared. So it's a product tool and the society is to explore for dx. So when you integrate you'll get excuse why is to explore five divided by five plus six. So this is something like a variable separable or converting to exact form. And suppose you all use the P of X. Q of X integrating factor. So we have differential equations divided by D X is equal to two x minus three, Y by X. So when you or transform into the standard form it is this and this is might be affects you affects is three by X. Keep our integration of three by access three log x. And if you use the properties of logarithms executes an integrating factor. So the solution will be by times of integrating factory integration of to Alex. Times of integrating factor plus C. And this is to expire four was integration is to expand five divided by five plus. See the same solution. Right? The fourth park, the fourth part is pretty simple. X plus, Y Dx is equal to, sorry, expressed by dx is going to X DY. So this can be original as X D y minus Y dx is equal to x dx. Oh, we can cleverly divide both sides by x square and this looks like a caution rule of why? By excursion? So what is the question to you by V. D O is the square? We do minus u t V. So this side is one by x. Dx now integrates Y by X is equal to log X plus C. And if you want to use P of x q of x matter, it's divided by dx is equal to one place of Y by X. That is D Y by D X minus Y by X is equal to one. Now Vfx is what? It's minus one bikes. And integrating factor will be par integral minus one bikes, which is the par minus log x. And that is one bikes. So the solution will be via times one bikes. This integration of cure fixes one, so one times integrating factories, one by X plus C. So why by X is equal to what is integration of this log marks plus six, That's it.

In mathematics a differential equation is an equation that relates for normal functions and their derivatives an initial problem but a problem or I. D. P. Is a differential equation along the appropriate number of initial conditions. So you have first everybody get vacations you over the X equals X. Sine X over Y. Can you stay operable techniques or we have, why do I call X M X dx shit. Then we integrate outside evacuation from the lap to the right side of vacation. X and X dx using the integral part. So it calls XB from SAn X. So we have negative X. Course and X minus the integral negative cause I'm X dx we have no that integral of negative causing X. Dx is equal to negative sine X. So we have equals two negative cause an X minus minus sine X. Simplify it. We have negative X. Cause an experience. I'm X add Afghanistan today solution. I got an ex course and experience an X plus C. So to that right up a condition we have Y squared over two. The integral. Why do I see sequence twice credible shit at this plus time plus sine X policy. So now we have Y squared over two equals negative X plus and X plus and X plus average quad three construct C or C. Over to So again across multiple better. So we're boycotters negative to expose and expressed his and express E. Re bend the initial value we have Y. Of zero is equal to negative one substituted to the solutions of differential equations. We have negative one calls negative two times zero causing zero plus two signs zero plus Z. So we have seats equals zero. We shall become zero. And also this 10 substitute today end the occupation here we have now. Why calls negative two X coffin expressed to us and X men's one

In this question. First of all, be able to fall our differential equation that is one minus x. Y. To the power minus to B X plus Y squared plus access. We're into one minus x Y to the power minus two. The white was 20 That's all week. First of all. And it is also provided when X equals to do what? It was 21 Now this is in the form of I can say that I am of the X plus and of Dy it was +20 Okay. And its solution is given by the formula that is integration of MDX less integration of N D Y. It was two costed while you are integrating them with respect to X. You have to do it here. Why is constant? Okay and in this integration here to take the terms there they are independent or tax. Okay. Now you see that we have to integrate here um that is one of only one minus x. Y. Holy square, Okay. D x And we have to take Y. S constant and the area to integrate the terms often that are independent of X and that is your only Y square. So we are to simply integrate Y squared Ok no we are going to solve this integration in the first integration. Your rotate, why is constant? So its integration should be minus one of 11 minus x Y 10 to minus one upon why plus integration of Y square is like you buy three who was to constant. So finally I can say that I get the value that is, it was too one upon. Why do I do it by one minus X. Y. That's why I Q by three equals to see. Now the condition is given to that is actually close to do and why it was 21 So I have to put it here that is X equals to two and why it goes to one. So if you want this condition here we can that the value of constant. I have to put weight was 21 so one of them one and 21 minus two. Bless one by three. Was to say these changes into seek was 21 by three and one minus four is minus one. So we got here minus one that is minus two by three. Okay, so finally I put this value up, see in this solution? So I got the solution of the question here. Okay, and the solution is one of one went into one minus X way less. Like you by three equals two minus two by three. Okay, this is the solution for the first question here and now you're brussels. The second question. Okay, so this was a solution for the first part and indeed seven part. We have to solve another differential equation and this differential equation I can say that is given to us. That is it was too well dx minus x. Dy it was too wide square the way. Okay, what is all this differential equations? I can say that first of all I am going to right here dividing by D Y. Then we got here. Why D X Y b y minus X. It was two Y squared. Ok, Now I divided by Y. So I get the X Y D y minus X by Y. It was too wide here. Now this is a linear differential equation. And to solve it we have to find first of all the integrating character. So the integrating factories, you do the power integration of the minus one of my own Y. Okay. Because this is in the form of the X. Y, dy less fever. It was two Q here. Okay, sorry the exit was took your so I take the values of P and Q. Here and after that I'm going to find the integrating factor. So integrating factories integration of he told the power to dx and here it is minus one of one Y. And its integration is to devour Eleanor, y minus Ln of Y. Now these changes into minus one of one way. Now I'm going to find the solution for this differential equation. The formula for solution is the solution comes out to be that is exciting here equals to integration. All you are you need explicit see okay, so you're delayed in the sea now I have to put the value of integrating factor that is minus one upon why it was the integration of Q. Q. His wife, so why into minus one upon right here. Okay and let's see now we have to integrate here. You can see that your way and one of one white cancellation and your minus X by Y. It was the integration of minus one with respect to leave. I should be minus wife. Let's see. So I got the solution for the second question. Okay this is a solution of the second question. Now we have to find the solution for the third part and the third question says that it is, it was too X into D Y by dx minus y minus five X to the power. Four minus three. Next cube sorry access square the last two weeks equals to zero. Okay now this can be lieutenant, divide by dx minus y by x equals two pilot skill and blessed three x minus two. Now to solo again we are to find the integrating sector here integrating factory. The integration of it to be about minus one of one years. So it should be into the bar minus one minus at the next hand after the installation of L m v D. You got your minus one of the annex. Okay now we have to find a solution for this differential equation and the formula for solution is why into a integrating sector. It was the integration of Q integrating sector. The explicit, see why do I face minus one of on it? It was too Q. Here is that is five X cubed last three x minus two. And your kiwis sorry I faced minus one of our necks B X. Let's see. Okay now we are solving this so we can say that this changes into minus y by x equals two. We divide all the times by minus X. So we get here minus five access were minus three. Listo by X. Okay, now I integrate all the time so I get minus Y by X. It was true minus five. Next killed by three minus three X. And plus to Elena affects. Let's see. Okay, this is the solution for the good question. Thank you.


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