what-are-the-integrating-factors-for-the-following-differential-equations-x-yprimexy

Question

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=x+y
$$

Prathan Jarupoonphol · Accepted Answer

in this video we are going to solve this differential equation Y. Prime minus why? Over a. Sorry? Y over X equals one. This is a standard form first we fi integrating factor P is -1 over X. So we integrated this and we get E. To the minus log of absolute eggs. This become one over absolute eggs, right? Yeah. Now we multiply both sides with this. And again we treat we can treat the absolute value as does no more polynomial because we have them on both sides. So when they are minors, they&#x27;re all minors. And so it just Like we have -1 multiply on both sides and it cancel each other out. So the left side here become why times 10. X. And this thing prime. Yeah Equal one over X. So if we integrate both side we have Y. Times one or eggs equals integrate this. The X. Is just love absolute X. Right Plus C. Yeah soul. Our function Y. Is X. Time long, absolute eggs plus see you bigs. And that is the answer. Thank you.

Prathan Jarupoonphol · Answer

high in this video. We are going to solve the equation. Why prime my That&#x27;s why equal Mmm to the eggs we are using Integrating factor as always. So fighting it is easy Do you? You just need to it Go p off x here wishes my last won the X So we get etude in mine ethics as integrating factor Once we found it be multiplied true, right? Both sides and the right side here gonna become just one And again the lips I can be as the by the ex off You did great. In fact, that time Why this always happen if you fire the correct integrating factors So we move the X to the right side and integrate the entire creation here we&#x27;re gonna have what e to the minus X y equals one plus some constant. Yeah, you can see that we put just one constant in because board off this integration, we have their own constant. But we just down them together into one term. We can do that. And so Oops. Sorry Mistake. There&#x27;s gonna be eggs plus c not one plus c so I&#x27;ll function. Why would b e to the X tons ace, plus the constant times into the eggs. And you can always shake it is correct by just plucking. Why? Back to our initial body and you&#x27;re gonna get the answer. That is it. Thank you.

Prathan Jarupoonphol · Answer

it&#x27;s real. We are going to solve this differential equation why prime minus Y over X. Equal minus lock eggs. So this is standard form. We first find integrating factor The P. Rx is -1 over X. So we integrate -1 over XDX. This is going to become minus a lot of absolute X. Right? Which cancer with E. L. Become one over X absolute eggs. So we multiply this true on both sides. And since we have absolute value on both sides we can treat it as normal polynomial. Because in the case that is is It appeared as -1 It&#x27;s essentially just having -1 multiply both side of the equation and it cancel each other out. So we treated as just polynomial. The left hands are going to become white times one over X. Of this prime. And now we just need to integrate the right hand side. And luckily this time even if we have block X here we can easily substitute. Why? As lock eggs. And sorry substitute us lock eggs. And do you gonna Far on this form one over X. DX which is exactly what we have. So by this substitution we have in the goal you D. You so we get new square Over two just like that now. Okay we have minus here. So is minus you square or what to plus a constant. And we just put this back and times X. To the right side to get the answer. Who&#x27;s gonna be X. Times oh minus lock X. Square over to plus yeah C times X. Right. Mhm. And that is it? This is the answer. Thank you.

Prathan Jarupoonphol · Answer

Hi. In this question, we are asked to find an integrating factor for this equation. Why prime equal X Y plus three. First thing in fighting integrating factor. You want to arrange your equation into the standard form that we use every time Here we do this to five p of X, and in this case it is minus X. When we help this, then we how if I&#x27;m alive for integrating factor as your ex equals e the constant e power to the p of it in the goal p m X dx Here it is usually a function. So it&#x27;s constant e power by this function. And let&#x27;s do this we integrate minus x the X and thing is just normal polynomial we would have yeah, minus X squared over two. Here. You usually have this constant term when we do indefinite in the call, right, But in this case, we can only them be careful just just for this integrating factor in this step that we can omit them in other other step, like usually solving, solving or the e u cannot omit this. Just be careful. Yeah, And that is it. In this question, we have integrating factor as sorry. We need to power. Even though this isn&#x27;t it, so is E two D minus X squared over two. This is our integrating factor. Thank you.

Robert Daugherty · Answer

section 4.5 problem 227. So we&#x27;re dealing with ordinary differential equations that can be written in standard forms. If I can write an equation like this, why prime plus some function of X y equals some function of X? Then I can integrate this using an integrating factor. So let&#x27;s get this in standard form. So it&#x27;s going to be why prime minus X y equal three x So the integrating factor is going to be e toothy integral minus X dx. So this is going to be e to the minus. X squared over two is the integrating factor. So I need to multiply both sides of this equation by the integrating factor. So when I do that, why prime minus X y equals three acts? I won&#x27;t buy all of that by e to the minus X squared over two. What you end up on the left side of the equation is just y e to the minus X squared over two. Prime is equal to three x e to the minus X squared over two, and then we can go ahead and look and say, Okay, what would it take them to integrate both sides of this equation. So when you integrate both sides of this equation on the left side of the equation, you&#x27;re going to get why e to the minus x squared over two? Uh huh. And then what we have on the right side of the equation? It might help if I write this as, um three. Then I could write it as, what, minus two x. So put a negative in front of their this and he to the minus. Excuse me? X squared over two. So if I did appreciate that um, good. Um, just minus X, sir. Let me back up. Sorry. Let&#x27;s do it like this. So I&#x27;m trying to do too many steps at one time. So what I can do here is make a substitution. If I let you equals e to the minus X squared over two, then do you is e to the minus. So that&#x27;s minus two X over to e to the minus. X. Um, pardon me. I&#x27;m every day, different day here. So if you differentiate that you&#x27;re going to get e to the minus X squared over two when you different shape minus x squared over to you get minus X. So what you see here is minus X dx and that in a row. So what this becomes is why e to the minus X squared over two is equal to minus three. And this is just the integral of e to the u. You did that you do you? So this is just minus three. He today you plus a constant of integration. So my answer here is why e to the minus x squared over two. The quota minus three you to the minus X squared over two plus a constant of integration. Now you divide everything by e to the minus x squared over to you get wise, you know, minus three plus c e to the X squared over two that is the family of functions that serve as the solution for this different equation.

Prathan Jarupoonphol · Answer

in this video. We are going to solve the difference, Joe Equation. Why prime minus two. Why equals minus X square by sewing, I mean, we going to fi their function. Why? What it is as a function of eggs. So it has to be something that satisfy this equation. Right? The way you can solve this, we are going to use integrating factors because this is first on the linear body. Um, we should know by this point that integrating factor is this into the integrating p of X wish here is minus two DX. So integrating factor is just yeah, to the my last two eggs. The way to solve this, you see, integrating factor, you will multiply. It is I will call it big, eh? The integrating factor. We multiply this throughout the equation on both sides. So you&#x27;re gonna get Yeah. Why? Prime minus to a Why equals my last ex square it Here is where the magic come from. The left Cy can be collapsed into X y prime. Sorry, A white prom, which is duty will be by the eggs. Why is that the kiss? This is not a coincidence. It is actually by these I we five this integrating factor in the first place just so that the left Cy when we multiply it true, the lifts I can collapse to this so you can try to work it out is gonna become this thing. You can just change room too. Find your away team office and it always works no matter what our body is that we start off. If you fight it, correct indicating factor. You always be able to collapse the lips I and the right side is still the same. So I gonna right out now. Sens This is just duty by the X. We can actually move in to the rice I and in the grid both sides. Why we do this? Well, you China, you eliminate Why prime here by changing things around and you end up with lips, eyes here, why? And something which is the result off the rice I hear. Now, if you cancel a out on both side by just multiplying them with inwards, then you&#x27;re gonna end up with why, on the rib side, on the left side, right equals to that thing on the right, and no matter what it is it is Ah, function off eggs. So here we&#x27;re gonna have our answer because we&#x27;re fighting. What function? Why should be to satisfy this equation and true this clearly the wise methought we and out with what? Why you look like so the content off body actually ends here. The trick is just this step. What left? He&#x27;s Can you Can we integrate this rice I And luckily, this time we can you see me go by part. Ah, So the rest off the video will be just fighting this You go buy pot. But I do tempt you to like try to understand this part off the main thought first so that you know what&#x27;s going on. Why were you We are doing this. The ego bypass is just technique. So you can practice that like separately. Now you go by pod. As I have mentioned before, you need just two thing. What is you and what is Devi? Everything else will follow from that. So the rights I you know the Minas I there. We&#x27;re gonna put it in later here. If you are bitten experienced, you&#x27;re gonna know right away that you should be X square and Devi should be the integrated infected in times DX. Why? If I had to give an answer, you could say that we won when we decide you, Andy, baby one d&#x27;you and V to be simpler or at least not getting more complicated. And here is the way to do is for this case give you instead, like so wish Place off X square and this exponential, you gonna have d&#x27;you as integrate the exponential right? And and we gonna get eggs Q or something over over tree, which make it harder in a way because the degree off eggs increase. So we don&#x27;t want that. That&#x27;s kind of the reasoning behind you can practice around with this technique, Really? And so I will actually skip through this. You just put put everything here. We have order, ingredient, right. Put it in and we&#x27;re gonna get X square. Well, my next to e. T. My last two eggs and here be last term will be this thing. When we like, cancel every constant and everything out. You can see that this is similar to the term we start with right but x square now become just eggs and you could say that is become simpler. So we&#x27;re on the right track. If we proceed with this the last term, we can either go by part again. You&#x27;re seeing similar are like technique, and we&#x27;re gonna end up with this ex Gru disappear now on the second term and I gonna right out here. You should try to doing this pod like alone later. Just for practice and discovery minus one over four off a menace to X minus. See? Do not forget the constant. This is what we get from from the term we started. Ah, this regarding the Minas I So when we pluck everything back, we&#x27;re gonna modify this my last true. Okay, Now, from from the start, the left side years integrating factor times why the right side use multiplied in my last. True, we can get X square or two integrating factor plus X over too integrating factor plus one over four Integrating factor and plus close. Then so you can see that this this e to the minus to a Q each other out except the last two. And we get a nice polynomial function. Oh, except except the last time there&#x27;s gonna be eat to the two X. Now, don&#x27;t confuse this with minus two. It&#x27;s because essentially we multiply both sides when e to the two X to conserve them out. So the last time I&#x27;m gonna have these attached to it and that is it. This is the general solution for our OD. You can always checked by plucking. Why back into how equation to see if it satisfy the equation or no, and that is it. What&#x27;s important is this technique at the start is important and it it&#x27;ll go by part is just a technique that you can practice. I hope this is helpful. Thank you.

Problem

What are the integrating factors for the followin…

Problem 228 Hard Difficulty

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=x+y
$$

Answer

$y=x \ln |x|+C x$

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$y=x \ln |x|+C x$

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Integration Review - Intro

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Catherine R.

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Kristen K.

Additional Mathematics Questions

Gilbert Strang,

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