5

C. y2 = 2yy" d. (D - 3)(D + l)y = l6x?e-= e A(c)y" (x) + A(c)y (w) + 43 =0...

Question

C. y2 = 2yy" d. (D - 3)(D + l)y = l6x?e-= e A(c)y" (x) + A(c)y (w) + 43 =0

C. y2 = 2yy" d. (D - 3)(D + l)y = l6x?e-= e A(c)y" (x) + A(c)y (w) + 43 =0



Answers

$$\frac{d y}{d x}+y^{3} x+y=0$$

In the question given system of equation that is three X plus two. White is equal to C. And four X plus five. Five is equal to be. Now in the question it is asking to find the solution of system by using in worst method so mhm. And in the kitchen he'd have two parts. So first I will find the value for part A means, find the solution. So in the occasion in part given that is metric subsidy is given that is equal to My takes off -11. So by holding in much method first, what we have to be frustrated. The medics equation that is X is equal to be. So this is metrics equation number one and second metrics are simplifying it. It can be equal to MS simple. Multiplying both sides inwards so X will become duties. It must be. So this is the second equation. Let assume I will use both equations one x 1. So as you see, A X is equal to be here. What are, what is the, what is the X. And what is the beast is it coefficient of X. Y. So do we redness. That is Back in the records. The elements are 3, 2, 45 and X. Isn't metrics of X. Y. And be easy metric subsidy and value is human for a party that is -11. It means metrics of -11. No. As you see what if Medic situation number two we have to find this a inwards. So now for finding in was I will use the formula because the medicine they had to close to order so we can get easily the value of inverse matics. By using palma dirties, one divided by a D minus B. C means three times of five -4 times of two. It will do that is 15 eight So it will become 1 7. And in the records, the elements are written as first flip the elements in the first technology means this is the first technol so It will become that is five and 3 then change the sign of element in the second diagnosed means this second diagnosed. So I do relate as managed to -4. So as you see, I had to get the inverse matrix. No use the second matrix situation. Which is saying that metrics of X. Sorry, X metrics means metrics of X. Y will be equal to mhm inverse B. So in Morsi's one divided by seven And in the records the womens are 5 -2 -4 and three. And metrics of B status, write a subsidy Those values that is -1, 1 soul. Now for next step, we help to simplify the metrics means multiplied the both metrics. So for multiplying I will use the dot product according to the dot product. What we have to do is we have to multiply row and column means this row and this column for finding first element. And for finding second element we have to multiple second row with first column so I can see that will be equal to metrics of X ray, will be equal to on the right side Here, I would write one divided by seven. Then in the record, the first element will be after multiplying First row with the column. So it will return is five times of one, managed to five times of -1 -2 times of one. No, for funding 2nd element of my takes Here, I will ride by multiplying second row to first column means -4 times of -1. This is 2nd row and this is the first column -4 times of -1 and Plus three times of one. No simplify this, then I will get metrics of X. Y. Is equal to one divided by seven bracket. In the bracket the first element will be returned yet After solving that is -5 -2, it will become -7. And here I will write minus minus minus become plus means plus four plus three means seven, No after no. Next step is missed, the elements on both sides. So value of X will be, I will get that is -7 divided by seven or I can say that -1 and value of what will be seven divided by seven. It all can say, I can say that, why will be one? So this is the solution of system even system by using inverse method No. For part B. Given that is it is given metrics of city that will be equal to Mathematics of 4, 3. So this strategy is given and we have to find the solution of system so use the second medic second equation that is saying X is equal to animals be and we have we have to find the we had get the n words in party whose value is status equal to One divided by seven. And in the records, the elements out that is 5 -2 -4 and three and no metrics we will be metric subsidy. So for part C. Sorry, part B the value is even that is 4, 3. So it means metrics B is equal to metrics of 43 Now use these metrics second equation. So I can say that Mathematics of sorry, matics actually charismatics of X. Y will be equal to in words a university that is one divided by seven. And the records, the elements are five minus two minus 43 And metrics be That will be written as matters of 4, 3. So in the next ship what we have to do is we have to multiply both. Matics means this by using blood products. So after multiplying what we get is metrics of X will be equal to When do I do by seven. Now for finding the first element multiply the first row, two column. Then it will region is five times of three. Sorry, if I time so far one is 2 times of three. And for finding second animated We have to multiple 2nd row two column. So with the matching elements. So it will return eyes -4 times of four Plus three times of three. No, after swallowing I will get the elements on the right side that is one divided by seven in the brackets. The elements are That is 20-6 means it becomes that is 14 And here I will write that is -16-plus 9. Yeah means minus seven. No national step is matching on both side so actually become deities. Yeah 14 divided by seven or I can say that that will be equal to two and why will be after matching on both sides. Value is that is 17 divided by seven. Or I can say that It will become -1. It means I had fine, I had get the value of X&Y. It means I had find I had got the solution of given equation. So this is the answer of given vision. Yeah. Thank you.

Okay, So for this system of OD, we are going to use the insemination method. So I'm we can be rightly or the, you know, for your way. So years an operator, the plaice for Aktuna, for us, the equation and the use Another operated B minus one. Acting on the second equation in the end, so we can use the 1st 1 Subtract the sick one, or the 2nd 1 step trickle for us. One the Alice Him. But because the second terms for both equation other Sam So you're free to the subjection. This term will be come. So again, we have B minus 11 X equals toe 40 prostitute. So this is the north from a genius or the so for the homogeneous part. That characteristic equation will be ar minus 11 equals zero. So I don't have one solution. I cost 11. That means the homogeneous solution for X will be see one times e to the eleventy and the further back particular solution. He's the right inside you, the mere function. So the particular solution will be another 1,000,000 function ap process. So we were going to figure out What's that? A wasabi by party in ex P back to the govt. So the minus 11 acting on XP because toe 40 plus two that gives us minus 11 a t press a minus 11 b you close to 40 plus two. Not by comparing the coefficient in front of T We have a close to minus for 11 and the P because to minors 10 years, six over 101. So, um, we have a complete general solution for X A C one you to eleventy minus 4 11 p miners to the six over 121. So this is X. As for why we can plug in, expect to one off this. Oh, these and the do and I know these over for why, But we don't have to do that in this case if I directly using equation one minus equation too. So Equation one minus the equation, too. It gives us for So we should use the question to minus he Question one. So we have four x plus five y equals to one minor. Steve. Basically, we can see that it's yr Libya dependent. So why? Because too, why Number five one minus T minus four or five X and we already know the eggs here. So just parking X into this equation so immediately we cannot 10. Why? So in the end, why coast of minus four or five? C one Italy eleventy plus one of 11 T plus 45 over 121. So you, in this case, C one is the same as the C one here.

Hello. The potion is taken from differential equation and we have to solve this differential equation. So differential equation is GQ plus three D. Spear -4 into wife. It's equal to zero. Since we cannot be equal to those of the group plus three D squared minus four is equal to zero. So let us use the central method like these equal to one initially. So we get one plus three minus four is equal to zero. So one is the first truth of the situation. Let us use the synthetic division in order to evaluate the equation after evaluating first group. So that will be 130- Book. So one went into one 44 into 144 into 140. So what the quadratic equation is? The square plus for the other equation is kind of quadratic equations. So that will be deep plus two holes where that is equal to zero. So other to sell the is equal to minus two minus two. So the solution of the situation will be why is equal to a plus B. X. into exponential to the power -2 waves plus. See exponential X. Which is the solution of the situation of this clears your doubt and

Okay, So to find critical points, we just need to set these equations for the X, d, t and D Y d t equal to zero. So just solving them as a system of equations. And when we set them equal to zero, we can get from this top equation that Y is equal to negative two X minus three. So we're gonna put that into the why and the second equation when we get negative. Three X minus two times negative two X minus three minus four is equal to zero. If we multiply that out, we end up with negative three X plus four X plus six minus four as equals zero, which gives us X equals negative too. So I'm gonna put that negative to back into this. The question for why, why is equal toe four minus three or just simply equal than one? So this system has exactly one critical point at negative to one. Okay? And so now that we have found the critical points we are to use So a software package um, yeah, okay. To see what's going on, I suggest doing that. So look better than this picture that I come up with but at you have Negative 21 is our critical point. If you look at the face plane, we'll see something that look like this. Okay? And so if we do as the textbook suggests and compare it to figure 5 12 we can see that this is going to be an unstable saddle point. Yeah.


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