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Question

Solve the following initial-value problems by using integrating factors.
$$
x^{2} y^{\prime}=x y-\ln x, y(1)=1
$$

Prathan Jarupoonphol · Accepted Answer

this video. We are going to solve this differential equation. Why? Prom minus y o X equals minus lock eggs over x square with the initial condition that when X is one, Why is one s will? So we have a rented into the standout form already. We know that p of X is minus one over eggs so we can find integrating factors as he to the minor locks. Absolute X wish is one over absolute X. Now, when we multiply this true on both sigh, we can consider the absolute value as just pulling a meal because we have absolute value on both side of the equation. In the case that give miners were Lou off them gonna give my nuts? Why, Lou? So we can cancel the minus one. How? From both sides So we can treated like this. Asked just no mobile e, no meal. Now, when we modified to the left side gonna collapse into this form as usual. What&#x27;s left is to integrate this thing on the right hand side. If we can integrate this, we gonna have that. Why use eggs times that thing that&#x27;d be integrated. So let&#x27;s do it here. I pull minus one out. We were inserted back later on. What we have to do is is obviously this style this home some take the eagle, buy pots used to sew everything. Um, so you will be long off eggs and then we will be one over X Q The ex. So do you will be. Why no eggs the eggs and we will be minus one over two x square. And so we have all the ingredients. We just put them in here. You we will be minus lock eggs over two x square And here we might ask to minus become plus one over two X cube now D X, And this is just pulling a meal so we can integrate it Fly away. It will become minus one over four X square. And don&#x27;t forget the Constanta. Yeah, I would like to write as minors so that when we multiply minus one, true this everything will become Plus, it doesn&#x27;t matter for for the constant is I can be whatever. Now what we have to do, we lock that in and we gonna get Yeah, I skip some steps. That is B I Q X through everything in here. We&#x27;re gonna get long X over two X plus one over four X says, see times, eggs. But it is not over. We have initial condition, Right? So x equal one, Gibbs. Why equal one as well? We have one equal lock off. One year, zero and this one over four. And this is C. So our constant room here will be tree or were four, right? And so we have. So this problem, we just inside this bag, this right here. So the answer to this particular equation is lock X over two eggs, plus one over four eggs, plus tree over four times eggs. And that is the answer. Thank you.

Prathan Jarupoonphol · Answer

Israel. We are going to solve this differential equation. Why? Prime minus two y equal x times e to the X with initial condition that X equals zero give Why equal minus one. Okay, um so it is stand out. Form P of X is just minus two. Integrating factor would be e to the minus two x We multiply this true left hand side going to become Why times e to the minus two x Oh, this primes equals right hand side will be here will be e to the minus X now And we have to integrate the right hands I So we use it to grow by past again. You is just x Do you is e to minus x d X Now the U is the X and we will be minus E to the minus eggs. Put everything in the formula we get, um to the end. It will be this. So this should be where if you should be very familiar with this technique already, and the term that you see Owen over like x time e to there&#x27;s something should be to it too hard. So practice. Ah, we put this in back in here, and I&#x27;m gonna skip to what? Why? Looks like here. So we shared from E to the minus to the, uh, plus X now, because we we used to have even a mite has two x on the left side. And like we calculate it out, we get this general solution. Now, we put in the condition here. Why is minus one when x is zero? So is zero minus one plus c. So the sea is actually Ciro this term gonna disappear and yeah, why will be just this, um, I thing You can pull this out. Yeah. Um sorry. Let me shake. Yes is correct. Uh, this is Thea Answer. There&#x27;s no constant yet because C is zero. That is the answer. Thank you.

Robert Daugherty · Answer

section Ford up five Problem. 245 Here we&#x27;re dealing with the near first order ordinary differential equations and I need to get this equation in standard form. So why prime plus some function of X times Why equal some other function of X? Then I will multiply by integrating factor which is just e to the anti derivative p of x. So to get this in standard form, this is weird to be why prime minus one over X squared plus one. Why is equal to negative one over X squared plus one? So my integrating factor is going to be e and then minus one over x squared plus one x the anti derivative there That&#x27;s just the arc tangent. So this is e to the minus our tension of X. So that is my integrating factor. So this equation reduces down to why times e to the minus arc tangent of X prime is equal to minus one x squared plus one and e to the minus arc tangent of X. Now it just becomes a matter of integrating both sides of this equation trivial. On the left side, it was the whole reason for a designing on this. So this is just going to be why and then e to the minus arc tangent of X. And then when I look at the right side of the equation can make a substitution here, let you, um, equal minus the art tangent of X Then d you going to remind us one over expert plus one DX. So in reality, what you see here with this particular integral, this is just simply the integral of e to the u Do you Ah, in that particular form. So this is going to be when you integrate that you&#x27;ll get e to the U plus a constant of integration. And so that&#x27;s just e to the minus our tension of X plus a constant of integration. So now, in order to solve for why you divide by e to the minus are tended of X, so you will get why is equal to So when I look at this So when I don&#x27;t fight this out, um, uh, you get y is equal to one plus see e to the arc tangent of X. And then I had an initial boundary condition. Wives zero is zero So why zero zero? That means that zero is equal to one plus. See e. The Arc Tangent of zero is just simply zero. So this means that C is equal to negative one. So my final answer is why is equal to one minus e to the arc Tangent of X. That&#x27;s the final answer here.

Prathan Jarupoonphol · Answer

this video. We are going to solve this equation. Why? Prime minus Why? What? X equal to lock eggs with the very condition X equal one give. Why&#x27;d you call five? So let&#x27;s do this. P of X is minus one over X That will give the integrating factor as her e to the minus lakh. Absolute ex wishes one over absolute eggs When we multiply this true both side. However, we can consider it ours and no more Polina meal. Because when it is miners, they&#x27;re all minors and the miners One can cancel each other out on this Gonna be X square. The left hand side will collapse too. Why? Times won over eggs prime. And what&#x27;s left is to integrate this term on the right hands. I And luckily, this time it is simple. We don&#x27;t need to use If you go into gold bypassed, We just use substitution lead you be locked eggs. So you buy the eggs will be one over X right? So the you will be won over eggs. Law. Sorry. The eggs wish Luckily is what we have here. Exactly. So we can translate this into and you go to you do you and we get you&#x27;re square constant just like that. That means we No, that right hand side in the grid into lot of X squared plus c. Now we plug it in here and multiply eggs. True. We&#x27;re gonna get this general solution for why we substitute dude X equal one and why you go If I were going to get this equation. Ah, love of Wendy&#x27;s zero. Right? So see is five. Now, the particular solution would be Hey, love off X square, plus five of six. And that is the answer. Thank you.

Prathan Jarupoonphol · Answer

Well, we are going to solve this differential equation. Why problem? And that&#x27;s why equals two X square with the initial audition that when x zero, why is zero. So this is in the standout form. We know that p of X is just minus one. So you get in fact, I will be into the in. The girl might just want the ex wishes e to remind us Eggs, right. We multiply this true both side of the equation. The left side gonna become Why? Times integrating factors Prime the right hands. I gonna be two x squared times e to the minus eggs. Now we have to integrate the right hands. I If we can do that, we&#x27;re going to get Why. So let&#x27;s do it first I pool number two out. We will insert it in later. We use our favorite techniques again. You go buy parts so you will be X square. Devi will be C to d minus x d eggs that makes do you, uh, two x, the x and we will be minus he to remind us eight putting them in formula. We get minus x square E to the minus eggs. This will become. Plus, if you grow two eggs either minus x d x and you can see that this is very similar to what we start with. Just chain X choir to eggs, right? We can do the illegal by part again. Yeah, uh, with, like Simula as Simon, we have you asked ex Andy we is the e part after this, you gonna have minus X squared E to the minus eggs, plus Sorry. Minus two x e to the minus. Eggs minus two X into the minus. X plus c So these are four. Turn from this. Now, don&#x27;t forget, we have two times two back in. So I just gonna move to the next page. And here from the original equation, we have that the integrated on the right hand side. Give us this term. So we just Monday pi e to the X into the eggs. Sorry. Okay. True both sides to eliminate them, and we&#x27;re gonna get that. Why is this this term here? Now? We still have initial condition to fire What? Where Lucy is. So when x zero, Why is zero it&#x27;s gonna all be zero left. My last four plus see? And so see will be just number four, right? And so we How did final answer? The particular solution would be minus to a square, minus four x minus full plus full E to the X. And that is the answer. Thank you.

Robert Daugherty · Answer

Section four without five problem to 49. We&#x27;re dealing with first order linear ordinary differential equations. The key to this equation. First, to get it in standard form, why prime plus a function of X times Why is equal to another function off X? So to place this one in standard form that&#x27;s going to be y prime minus one over root X Why is equal to two X divided by the square root of acts? So that&#x27;s just gonna be too square root of X. So the integrating factor is going to be e to the minus one over root X d x. And so this is going to be e tu minus two root X So that&#x27;s my integrating factor. So this equation turns into why e to the minus two root X prime is equal to to root x you to the minus two root x. Now, then the key becomes integrating both sides of this equation. So to integrate both sides of this equation on the left, I get why e tu minus two root X is equal to know when I integrate on the right side, I&#x27;m gonna end up with two x plus two x plus one times E to the minus two root X plus a constant of integration. Now I will solve for wine by dividing by E to the minus two root x and that leave me with minus two x plus to root X plus one. So all of this plus see you to the to root X. You have the boundary condition that y zero is one. So that would give me that one is equal to minus zero plus zero plus one plus c and then e to the zero is just going to be one there. So this tells me that one is equal to minus one plus C. Therefore, C is equal to two. So this final answer is why is equal to minus two X plus to root X plus one and then what states use it with the two plus two e to the to root X, and that is our final answer.

Problem

Solve the following initial-value problems by usi…

Problem 244 Hard Difficulty

Solve the following initial-value problems by using integrating factors.
$$
x^{2} y^{\prime}=x y-\ln x, y(1)=1
$$

Answer

$y=\frac{\ln (x)}{2 x}+\frac{1}{4 x}+\frac{3}{4} x$

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$y=\frac{\ln (x)}{2 x}+\frac{1}{4 x}+\frac{3}{4} x$

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Caleb E.

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Calculus