what-are-the-integrating-factors-for-the-following-differential-equations-yprime3-xx-y

Question

What are the integrating factors for the following differential equations?
$$
y^{\prime}=3 x+x y
$$

Robert Daugherty · Accepted 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

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.

Prathan Jarupoonphol · Answer

we are going to solve this differential equation. Why prom minus Y over X plus two equal three X over X plus two. This is the standard form. So first step we gonna find integrating factors Here P of X is -1 over X plus two. Right? So we integrate this the eggs and power E. To this amount we&#x27;re going to get minus Long, absolute x plus two over eat. And this is lock is on base E. Right? So we can cancel it out This become one over Absolute X-plus two now. Mhm. The absolute value may looks complicated at first but in this case it can be easily there with because we are multiplying both sides of the equation with this amount. So yeah, you&#x27;re gonna see in a minute why is not a big deal when we first multiply this true, we&#x27;re going to have absolute value left over like this. But then when you consider all the cases possible when when absolutely a little give plus on minus side you&#x27;re going to see that on a case where they are minus is essentially our original equation multiply -1 on both sides. Right? That means this -1, cancel each other out in the end. So no matter what value of X is we can always make it so that we get the positive side because minus one can be canceled on both sides. This is going to become just pulling a meal X plus two square. And so we can easily stop this. So left side gonna collapsed too. Yes, prime. So why times X plus two, prime equals three X over X plus two square. And now if we can integrate the right hand side, we&#x27;re gonna help. Why? So let&#x27;s do it. This is also not complicated. We use substitution let you be X plus two then the you is just the X. Right? So we share this too in the grade three export B tree of you minus two. Right? Or were you square? Yeah and Kia D. You So this is polynomial. We can Break it into two parts tree over you minus uh six over your square. All right. And both of them just holding the meal so we get three love absolute you minus This will be plus six over you. Right? Because integrated when we use choir We get -1. So this is the right hand side And you is just X-plus two here. So now we can go back and see that all we have to do is multiply X plus two two. That amount that we just get and the answer will be this value. Mhm Don&#x27;t forget the constant term here. I think I forget to write it down. Okay do not forget the constant term here. And that is it. This is absolutely X plus two. And this is the answer. Thank you

Robert Daugherty · Answer

four, not 500 to 31. We&#x27;re dealing with ordinary linear first order, different two equations. So in this case, the standard form is going to be why prime plus a function of X times Why is equal to another function of X? So if I can get it in that form, I know that my integrating factor is going to be easy to the anti derivative of P. So to put this in standard form, you&#x27;ll have why prime minus three over X Y is equal to X. And so what you need now is the integrating factor. Mu of X is equal to e to the minus three over X the X So this is you to the minus three natural log of X E to the natural log of X to the minus. Three is extra the minus three. So that tells me that this equation can now be written. Um, right here this equation can be written as the integrating factor is X cube. So why, over X? Cubed crime is equal to x times one over x cubed. So this tells me that why over X cubed prime is equal to one over x squared, and then I would integrate both sides of this equation. So when you integrate this, you end up with y over X cubed that when you integrate this, this is gonna be what, minus one over X plus a constant. And then now we just simply sought for why Why is equal to X cubes as we minus X squared plus c x cubed? So that is the solution to this different equation.

Prathan Jarupoonphol · Answer

fine in this video, we are going to fi and integrating factor off this equation. Why Prime equals X law. Thanks. Why plus three of its so first thing we arrange it into the standard form for linear equation. You have X lock X Y on the left side, entry of ex dare. So here I will pee off eggs is minus X lockets and integrating factor is how well plover eggs integrate p of x d eggs. So if we can find this me, we will know the denigrating factor. Now many of you will have come to this conclusion come to this step and stuck because off this function X lock X is not is not seem bo function to integrate because you cannot use like you show. I didn&#x27;t did these all like function to solve these. There&#x27;s no, like simbolo functioned directly by the X become this thing is more complicated than that. So how do you solve this? You can use integration bypassed and to go buy cards. And for those of you who who are not familiar with this term, what it is is I think of it as a way to break a complicated integration into a symbol. Pieces. Um I do write the formal lad out first in say that in the goal you Devi equals you be by less if you go, We did you This may seem confusing. It is the first time you see this, but it is very useful too. And you are going to use it always over in this cost. So please try to look it up and be familiar with it. One we have to do to use Indigo bypass is the to my What is you at? What is Devi from from the term we want to integrated, then everything else will follow. We have to decide this to first. So let&#x27;s do this Our term. Years in the gold, eggs, knock eggs, the eggs. Oh, first thing first I signal in the minors. Like from this point on what? We&#x27;re gonna put it in at the end. So just annoyed for now, just for convenience sake. Um So as I said, what is you? What is DVD? First thing you need to know is the Devi pot will always always have the X in it. So, off the three things here, the ex already? Yeah, in dv pot, we have to do to my what you&#x27;ll be. It can be eggs, a lot of eggs or both of them together. But if issues the holding a Glock eggs, we&#x27;re gonna come back to the initial question and there&#x27;s no simply quit simplification there. So we cannot use that every use eggs. He oh, gonna have law eggs, which is very complicated as well, because we cannot simply integrate this to get V. We decided the O. V to get B, and this is not easily solvable. We have to in wall and not the indigo by path. So this is not appropriate either. In this case, we have you asked a lot of ex alone and the we asked x the X That is the simplest way to do it. Now we I&#x27;m gonna write this doll again. Uh, you re minus you go be Do you is what we want And he&#x27;s gonna geek woes equals our in the show terms. So you&#x27;d be happy ex the ex Ask Devi what is really we simply we simply integrate both sigh. So we put in the girl here and here and Nicole DVD gonna be free to call x d X we know is X square over too. And here, the constant tomb off this process all fighting integrating factor We can omit the constant term But be beware. That is only in this step If you fi the general solution all for the e all other times you integrate Really? You sure not forget the constant er Okay, now we have Do we have we what we lack? The last thing is Devi, right? Sorry. The U and we can do this by simply find the deputy Do you by the X Since you is locked off next week Me We have the lock picks with the ex wish If you know from the cause material is gonna be one over X And so the you alone can be written asked the eggs Oh, eggs And we have all the ingredients need to feel this formula So here we go You is lock X We&#x27;II is X square over too Mine us integral V E&#x27;s X squared over two and the u is the ex. Well, thanks. You can see that here We have to go All the eggs off function concerning eggs. So everything is fine and the first term years done already. So we leave it like that, the second term here, x square, and it&#x27;s gonna can so one order out and is become integrated X o into the mix. And you&#x27;re gonna get X square or were X squared over four. Right, Because we we have one over to appear from the integration, and this is it. This is equal to our initial in the show terms, and we can go back and say that. Okay. Our integrating factor is mmm. Power bi by this right, which we just Okay, it&#x27;s my last this my last days. So he&#x27;s mine. Us. Um look, X X square. Well, went to minus X square. Louisville. Okay. Sorry for these clams Space. So hee constant e power by minus. This this whole terms is our integrating factors. You can, like, multiply minus wanted. And she and the this eye, but is the same. So this is the answer. Thank you. Sorry for the long video, but this is like a new new technique that I introduced integrating buy into gold by heart. So I want everyone to I understand

Prathan Jarupoonphol · 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.

Problem

What are the integrating factors for the followin…

Problem 227 Hard Difficulty

What are the integrating factors for the following differential equations?
$$
y^{\prime}=3 x+x y
$$

Answer

$y=-3+C e^{\frac{x^{2}}{2}}$

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Gilbert Strang,

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Integration Review - Intro

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$y=-3+C e^{\frac{x^{2}}{2}}$

Calculus

Grace H.

Catherine R.

Anna Marie V.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Gilbert Strang,Calculus

Grace H.

Catherine R.

Anna Marie V.

Joseph L.

Additional Mathematics Questions

Gilbert Strang,

Calculus