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(25,00 Puanlar) Which of the following differential equations has solutions of the given curve family % + cy? = 17 a)z?y' +y+ 1 = b)wy+ (1 _ :)y =0 cJoy cy = 1...

Question

(25,00 Puanlar) Which of the following differential equations has solutions of the given curve family % + cy? = 17 a)z?y' +y+ 1 = b)wy+ (1 _ :)y =0 cJoy cy = 1 d)zly+ (1 r)ya) 0 a) b) b} c) d) Bos birak<Onceki3/4Sonraki>KapatVSinavi Bitir

(25,00 Puanlar) Which of the following differential equations has solutions of the given curve family % + cy? = 17 a)z?y' +y+ 1 = b)wy+ (1 _ :)y =0 cJoy cy = 1 d)zly+ (1 r)y a) 0 a) b) b} c) d) Bos birak <Onceki 3/4 Sonraki> Kapat VSinavi Bitir



Answers

$15-18$ (a) Find the differential $d y$ and (b) evaluate $d y$ for the given values of $x$ and $d x .$
$$y=\frac{x+1}{x-1}, \quad x=2, \quad d x=0.05$$

Were given a function and in part a were asked to find the differential of this function. Function is why equals E to the X over tea? Sorry, x over 10. So we have. If we call this ffx, then F prime of X is equal to buy the chain rule 1/10 e to the X over 10 and therefore the differential d Y is equal to e to the X Over 10 over 10 DX in part B were given values of X and DX and we were asked to evaluate the differential for these values were given that X is equal to zero and the D X is equal to 0.1 then from part A, we had the differential de y is going to be 1/10 of E to the 0/10 times 0.1 and this is simplified 2.1

The institutions. We have to find the differential equation of the family of curl. That is why is equals two into into the power three X. Plus B. Into into the power five facts. And and we are the arbitrary constant. So I am going to differentiate the situation so different shoot with respect to X. And we get This is the way by DX is equals two 382 E. To the power three X. Plus. This is five B. Into into the power Fairfax. Let this is our equation 2nd. And this is the only question 1st. Now I'm going to differentiate it again. So this is the two white by the excess square as equals to nine into the part three X. Plus. This is 25 into B. To the power five X. That this is our equation third. Now I'm going to subject equation three minus aggression too. So this will comes out to be The two. Bye bye. The excess square minus. I'm going to multiply it with five also. So this will comes out to be five divided by D. X. S equals two minus six A. To the Power three X. Now from here we get the value of A. Into E. To the power three X. That is minus one by six. Into the to abide by D X squared minus five, divide by dx. No, similarly I am going to subject equation 3 -3 aggression. And this is multiplied by three. So this film comes out to be The two by by be excess where -3 into divide by DX. And this is then into mm sorry 10 B into the power five fax. From here we get the value of being too into the power five fax. That is one by 10 data. Wait by the exit square minus. Trying to divide by dx. Now for these values in equation one. So from the equation when we get this is why is equals two minus one by six into the two by by the X. is where Plus this is five x 16 to divide by the X. Plus one x 10 into day too. Bye bye. The axis where -3 x 10 into the Vibe, I the X. Now I can simplify this and this with me. The return is do you do right by the exit square -8 into divide by DX Plus 15 by equals to zero. So this is the require differential equation of the family of God. And this is our answer for the institution and for that option is the correct answer. Thank you.

All right, it's It's over. Problem 51. He had to find the solution to this differential equation and then graphic. So when you have a simple differential equation like when older coefficients are constants, then you can assume that the answer is in the form of need to the time for constant times the independent variable which will use X for So we just arrived This So why are private secret que times each of the chaos collectible products, he writes a k a score of times each of the King X and there We substitute these into the differential equation. So it's going to be caseworker and each of the King X plus five K each of the K X plus 15 he to the King X is equal to zero. Be a factor on each of the chaos It was gonna be each of the text times Case group was five k plus 15 0 to 0. We know that need to the K axis never gonna equal to zero, since it's an exponential function, which means we're gonna have to rely on the rights party equals a zero and conveniently is just a quadratic equation case curry plus five K plus 15 secret zero. And it doesn't seem to be fact herbal. So we'll use the quadratic formula I need to find customize a square for throwing lines four times one times 15 all over. Two times one thanks five plus or minus the square root of 25 minus 60/2. Can you the five person lines the squares of negative 35 all over itself. This skirt of negative 35 is just i times roots 35. So in the end, our solutions for KR Casey goes negative. Five plus remind was I routes 35/2. Now, we're gonna use a shortcut. Teoh directly find a solution to the differential equation. And there is a more detailed way to solve this in some of the other videos. Book. We're not gonna do that here, Uh, just so we could save up on time. So in the solutions for KR, in the form of Casey goes an Alfa poster minus days I and you can immediately find the general solution for the different equation that goes as follows. Years of hard Alfa times. Next times a constant times coastline of pay the ax, plus another constant time sign of Vega X. So now we just deployed in values. So we're gonna have to split apart this fraction for us. So it's gonna be negative. Five forward to plus or minus. I rode 35 over it. So So in this case, our Alfa is gonna be negative 500. So and our veda is gonna be roots 35 over its here and we just to put these into our four months so it's gonna be each of the negative five x over two times. See one times Co Simon, Route 35 Words here. Next for C two times, sign off. It's 35/2 X And just like that, we found the general solution to our differential equation. So it's gonna be why equals all of this? So now we just a plug in values to find the two unknown constants. So when Access zero why is negative? So So why is giving negative Tosu years of the car off zero, which is just one time see, once I was co sign of Route three, rich 35 or two times zero is zero and then force seizures I'm sign of zero. So e to the power zero is just one so we can ignore that sign of zero shows. Zero. So we can Nora, that's and their co signing. Zero start the one so we can ignore that and those a c 10 turn negative. And then we have the driver to find the other initial to plug in the other initial values. So when we derive that, if we're going to get negative 5/2 times E to the negative five extra or two times, we'll see one. That's negative, too. So it's gonna be negative two times roof Ruth 35 or two. So it's gonna be name. There were 35 words. So and since there's a co signs gonna negates once again so it's gonna cause it's a recertify. It's time. Sign of through 35/2 acts and then for us Route 35 to see two times Night. We already have a sign co sign of we were 35 words. So acts Wait. I think I derived a both forest. The simultaneous sleep. Okay, so I'm just gonna you move this down here. And so when we drive the first time. We're only going to derive like this park. And when we drive us for the second time, we're only gonna derive this part. So for the first time, we're gonna rewrites the entire second part. So it's gonna be *** two times, cause I 35 words x First theater time signer through 35 over its do eggs plus e to the negative five x over two and then times all of this. So now that we've actually gotten the derivative, let's play gin Italians So in excess zero Why prime nous seven So seven skin equal Teoh sons Negative five House times e to the zeroth Power times near two times. Cool sign of zero. Sorry. Zero Plus the two time Scient zero plus you need to the zero of Power times, Route 35. Time sign of his zero plus the roots 35 or two See to times co sign of zero Son of zeros. Just zero. So we can, you know, learn Thoenes. Each of the first powers just want each of the zeroth promise of one. So the kings north and co sign zero is just one, so I can ignore these. So in the end, we've got something is you go to a negative five house times. Negative. So plus Route 35/2. Time seat and yeah, we just try the isolate C two. So is you mean seven. Segal's five plus were three highlights here. See it. It's attractive five from both sides. We get to go to through 300 to see to and then divide both sides. By the way, we don't have, you know. Well, that's okay. We just drive. Divide both sides by route 35 were too. So more guts seed service. See through theater four over Roots 35. And we're gonna have the race on stuffs. And where are the room They are. Final solution is gonna be, uh where do I write this? No, I just race the first parts. So our final solution to this difference in equation along with this initial values are yes. Why? People Teoh years of ah, negative. Five acts over two times negative zero times co signer Route 35 over. Two acts plus 4/35 time. Sign of her 35 over to axe. And if you were, it's a graph is gonna look something like the We have a bunch of almost vertical lines on the left, and then you get like this and, yeah, that's basically elements.

In part a were given a function were asked to find the differential were given that why is equal to the square root of three plus X squared. Call this ffx than F prime of X is equal to buy the chain rule when half times three plus x squared the negative one half times two x, which simplifies two x over square root of three plus X squared and therefore the differential D Y is equal to X over the square root of three plus X squared times. DX. It's in part B were given the values of X and DX and were asked to evaluate differential. We're told that X is equal to one and the D X is equal to negative 0.1. Therefore, by the formula from party differential, D Y is equal to one over the square root of three plus one times negative 0.1 simplifying we get one half times negative 10.1, which is negative point 05


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