what-are-the-integrating-factors-for-the-following-differential-equations-x-yprime3-y-6-x2

Question

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=3 y-6 x^{2}
$$

Robert Daugherty · Accepted Answer

Section four Doubt five problem 225. We have a first order linear ordinary differential equation. OK, The standard form for this is why prime plus some function of X times Why equal some other function of X And then the integrating factor is going to be e and the anti derivative of P of x dx. So to get this in standard form, you&#x27;re gonna have X Y prime minus three. Why equal minus six x squared. Divide everything by X. Why prime minus three over X. Why is equal to minus six X So that&#x27;s standard form. And then my integrating factor is going to be e so the anti derivative of minus three over X dx. So this is E to the minus three natural log of X that is e to the natural log of X to the minus three. And so that is simply X to the minus three power. Yeah, So I need to take this equation which is in standard form and multiply both sides by one over X cubed. Okay, So when you multiply the entire equation, you&#x27;re gonna have why times one over X cubed prime is equal to minus six. X time is one over execute. So this is why over X Cubed prime is equal to minus six over X squared. And now we are able to integrate both sides with respect to X. And we end up with why over X cubed it&#x27;s equal to when you integrate this grunting end up with six over X plus a constant. Now you multiply everything on both sides of the equation by X cubed in order to solve for why should I get why is equal to six x squared plus c x cubed that is the family of solutions for this ordinary different your question.

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

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

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

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 225 Hard Difficulty

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=3 y-6 x^{2}
$$

Answer

$y=6 x^{2}+C x^{3}$

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