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

$y=\frac{x^{2}}{2}+\frac{x}{2}+\frac{1}{4}+C e^{2 x}$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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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'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'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're gonna have our answer because we'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'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'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're gonna put it in later here. If you are bitten experienced, you'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'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'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't want that. That'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'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're on the right track. If we proceed with this the last term, we can either go by part again. You're seeing similar are like technique, and we'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'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's gonna be eat to the two X. Now, don't confuse this with minus two. It's because essentially we multiply both sides when e to the two X to conserve them out. So the last time I'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's important is this technique at the start is important and it it'll go by part is just a technique that you can practice. I hope this is helpful. Thank you.

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