5

(Hint: Use the differential of A to answer this questions )S203.22,S446.45,S669.67S212.22, S428.45, S642.675212.22,S426.45,S642.67S213.22, S446.45,S639.67S213.22, 5...

Question

(Hint: Use the differential of A to answer this questions )S203.22,S446.45,S669.67S212.22, S428.45, S642.675212.22,S426.45,S642.67S213.22, S446.45,S639.67S213.22, 5426.45, S639.67

(Hint: Use the differential of A to answer this questions ) S203.22,S446.45,S669.67 S212.22, S428.45, S642.67 5212.22,S426.45,S642.67 S213.22, S446.45,S639.67 S213.22, 5426.45, S639.67



Answers

Use the technique derived in the previous problem to solve the given differential equation. $$x y^{\prime}-y=x^{2} \ln x$$

Hello. Let's get this ominous problem, shall we? So, step one. We need to get this equation distended forms so that we can use this equation. So the standard form is I would get a divide. Everything by Excellent. Why prime minus one over X. Why is equal to X allen of X? There we go. Because excreta bit of excess x so now are going to do is plug and chug for reference. This will be small p of X, and this will be a queue of X. So now, to solve this equation, we need big P of X. Big P of X is going to be equal to the integral of small P of X. So in this case is what is well in front of the why which is going to be the integral of negative one over X, which is going to be negative. Ellen of X. So we have instances where eat a p of X and each of the miners your backs. So we in the either miners p of X is going to equal to e to the minus of minus Ln x, which is going to equal to eat to the Ellen of X, which, if you do eat it, Ellen of X, you pull up what's in front of the l innuendo, but acts e to the p of X. You just end up having e to the minus. L of X. A minus sign goes up top here. So you end up with E to the Ellen off extra minus one, which you just again plug you take a what's in front of Ln which will be won over axe. Okay, so now we're replaces appropriate. So we end up with why off X is equal to the one X plus the integral of ah, one of her ex tat. This would be one over x So one over x times x times l a necks So you know, But one of the x times x, which is going to be just you end up with Ellen of X and all this times X there we go. So interested in a row LX today, Digger integral elements we use use equal to Alan mx and we ended with DVB equal to one on the former is going to be u times v minus integral sign of v times Do you. So we end up with C one of X close U times V, which is going to be equal to X times Ellen of X, because integral 01 is X minus Integral science There were two of you is going to be equal to one of her acts. Times integral of excellent be Interval one x and all of this Times X. So the one of her X and X will cancel out to give us just once we did the interview that one again meeting on the outside Here we end up with C one of X plus x times pellet of X on an interim of one is going to be X and all that time.

All right, Tonto. Do Lena works emission off for candor. Course you can't. Siegen perplex examples. I do. Uh, so this lean approximation Well, it's a line. Um, since it close the deal, that's cool. You know else to do so these linear approximation for a function. Well, this is our function. Here are fortunate. Factories go see, can't fix works. So the approximation to be able to There are any of the fun not to plus the body off day diverted two times, huh? X so for far apart from two. Your ex, My celebrity friends It's the accent Do also for dysfunction. This is the best near Brooks emission. Let's go points. You know, like that was a Have some point here seem that this is your point two because he kind of too. They were never ex mention Is that lying? But the student man do touch that point. So? So the linear approximation is the line. That much is that point? Uh, you stay close enough to their points. Do you two should be here in the X axis. It is that you stayed close enough to go the linear approximation. Staying on that line is not be so far apart from regardless of the function. So, uh, or dysfunction. What is What is a crime? What is native? You eventually called C. Thanks for the people that it's my house. Cold dungeon off Rex terms. Cosi pontifex. Yes, you. So all these things that works commission will be even by this particular one. But we're looking at would be what? Oh, two plus. Well, here comes the miners. So miners called tangent off too. Can schools two? Those are numbers. Well, numbers in the times X minus two. Not that, however, finds it is. My look, those just Those are just numbers so that peace will be no next. He's just a light. Well, probably. Dad, Probably. The function is not like that, but, uh, but that is idea that line the approximate CE fortune that the land based approximates I think you should leave this approximation year four points close enough to do so. Uh, Asan example. We have you here on this, least for X. Where is the money off, coz she goingto Becks. So you've got these X opposite bugging Cosi pump the brakes years about linear approximation to so as you can see from this table is the's an approximation, he said. Threat to the linear approximation agrees they function to so airports. Michelle too, Which is? This value is equal to the body of the function. Two from the formula. We can see that it's a little girls you plug in X equals doing here you have here to us through bang that's gone. So this piece would be gone yourself cause he went to on a CZ, you can see well, the the air there starts getting. Why no wider as you go far apart. So here here is not so, but there is more. But your memory is not so bad. He is. We could get wars. So yes, all the values on the stable will be said, uh, computing because he went off to minus pageant of two them school sequins off too. Bombs, Eggs. What this for these particular X here. All these this

Let's find the differential of the given function and here noticed that l We can rewrite this as El of X. Why's he these? There are variables so that the formula deal That's a differential of El. So first, we'LL have the differential of l with respect to X and then multiply that by the x derivative of El with her Spencer. Why times d y? And don't forget the third variable here Z so partial derivative of El with respect, izzie And then times easy. So we'LL need to go ahead and compute these three quantities thes partial derivatives. So first one derivative with respect to X So we just differentiate that that goes away to a one and we're left over with z e negative y squared minus x Excuse me minus C square so we can go ahead and plug that in for the first part. So it's a minus Z square and then don't forget the DX. So next let's go to the partial derivative of El. With respect, sir. Why So this time when we differentiate, we have a Y in the exponents. So then we'LL have to use the chain rule to differentiate the exponents with respect, sir. Why? So we get a negative, too way out of that. So let's write. This is minus two X y z and then e negative y square minus Z squared. And don't forget the D y. And we have one more term to go here L C D Z. And this time we will need the product rule because we see easy appearing and two different factors ones here and then also in the exponents. So use the product rule there she different. She's easy. It just goes toe one and then for the next term when we differentiate, will still have our eggs e. But when we differentiate the exponents using the chain rule, we'LL get a negative to Z. So there's the negative, too. And then the additional factor of ze gives us a Z square there, so this will be minus two x z square e negative y square minus C square. We've been very careful here and include the DC and there's our final answer. So D'Oh and these expressions on the right give us the differential of El. That's our final answer

Hello. Let's get the summons problem, shall we? So we need a p of X and the cure vexed in particular. We need this big p of X on disk. You, Becks. Luckily, Q of X. It's going to be right here. Unfortunately, we have p of X, which is our small p of X is gonna be right in front of our Why, in theory, what's in front or y in this case is just one. So if you want big P of X, we need to take the integral of small Pia Banks. So in this case, we said smoke P of X is just one. So we didn't integral one, which is just X. So now, in this case, in his problem, we have e to the big p of X to the not case is just going to be easy. The X and in other cases we have eating a minus p of X. So in that case is just going to be eating a minus X. There we go. Now, would you need the plug and chug as we need? Why of X is equal to see one eat the minus X there ago, plus into grow of cure vex, which in this case, is gonna b e to the minus two x e to the minus two x times p of x et et Pierret vexed with me e to the X and all of that times e to the minus X There we go. Nothing fancy at all, really? So hope here we're gonna end up with X minus two of X Where you going to give us a minus? Acts? There we go. And then finally we end up having to the integral that bullets right on. So we see one e to the minus x plus integral of e to the minus X and all that times E to the minus X which is going to be equal to C one e to the minus X plus the integral of each of the mind. Sex is going to be the minus X, and we take whatever concert and divided by that concept. So in this case, it's minus one, so it's minus one times E to the minus X. In this case, the exponential is will act together. So we end up with C one e to the minus X minus e to the minus to Iraq


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