what-are-the-integrating-factors-for-the-following-differential-equations-fracd-yd-xtanh-x-y1

Question

What are the integrating factors for the following differential equations?
$$
\frac{d y}{d x}=	anh (x) y+1
$$

Robert Daugherty · Accepted Answer

45 Problem number 2 21 were asked to find the integrating factor for this first order linear, ordinary differential equation. So that form is going to be D Y d X plus some function of X times y is equal to some function of X. So to get it in that form, I&#x27;m gonna have what d y d x minus the hyperbolic tangent of axe times Why is equal to one. So now you can see that p of X is this minus hyperbolic tangent of X. So my integrating factor is going to be e to the integral of p of x, the axe, the anti derivative. So this is E and in the anti derivative of minus hyperbolic tangent of X, the X And now we can go through a little bit of a simplification, the anti derivative of minus hyperbolic tangent That&#x27;s minus log Hyperbolic co sign. So this is going to be e to the minus natural log of the hyperbolic co sign of X. Okay, so this is e to the natural log, um, of the hyperbolic co sign of X to the minus one, continuing with our property of exponents. This is e to the natural log of one over the hyperbolic CO sign of X and then e to the natural log of anything. So this answer is going to be one over hyperbolic co sign X. So that is the integrating factor for this particular linear, ordinary, different equation.

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.

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

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

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

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.

Problem

What are the integrating factors for the followin…

Problem 221 Medium Difficulty

What are the integrating factors for the following differential equations?
$$
\frac{d y}{d x}=\tanh (x) y+1
$$

Answer

$\mu(x)=\frac{1}{\cosh (x)}$

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Gilbert Strang,

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$\mu(x)=\frac{1}{\cosh (x)}$

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Caleb E.

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Integration Review - Intro

Definite vs Indefinite

Gilbert Strang,Calculus

Heather Z.

Caleb E.

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Samuel H.

Additional Mathematics Questions

Gilbert Strang,

Calculus