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Consider the following differential equation: 4y' = 26y . Which of the following is an integrating factor for this DE?exp 2+0)expexp28exp28+C...

Question

Consider the following differential equation: 4y' = 26y . Which of the following is an integrating factor for this DE?exp 2+0)expexp28exp28+C

Consider the following differential equation: 4y' = 26y . Which of the following is an integrating factor for this DE? exp 2+0) exp exp 28 exp 28 +C



Answers

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

So to start off this problem I'm gonna go ahead and move the 72 the X. To the other side. So what we'll have is this equals to seven ft of the X. All right. I'm gonna rewrite why? Prime as dy dx The sequence of 78 of the X. Okay. And I'm gonna go ahead and move the dx the other side. Mhm. Okay. I can integrate both sides. Yes. So on the left hand side that's going to be six Y squared. And the right hand side is just going to be seven E. To the X. Plus E. I'm gonna go ahead and divide by six so Y squared equals 276 E. To the X. Policy. And I do not want to take the screws so I'm just gonna leave my answer like this so that's my fault.

So for this one, or just sort of Ah, just like pick the pace up. I just kind of wrote everything down already. So getting into the lesson we have em, which is going to be equal to why and what's going to be equal to negative two x minus for race. The why I raised the fourth power and when we take the derivatives of them and and with the and with the derivative of why is gonna be one and end with the derivative of X is gonna be negative too. So what we're gonna do is that we're going to take em y and attracted by an ex. And now, in order to determine if it's we're going to divide by negative M or N re, it's going to be by M. And the reason why is because r m, why are when we subtract R M y bar and next, we're going to get a constant and our, um, and it and the only one and the only one between the and between, like the M and the ends with the M, is going to be the only one that has a one variable. So when we So now that we picked there and we're going to get three over negative? Why? So when we have ah, plug this into R E and we tried to, um and we try to sort of take its integral. What we do is that we get one over. Why cubed. Ah, and that's going to be our, um What was it, Kolya? Our integration factor. And so for our, um, em over Aziz when we plugged, When we multiply our M buyer, I we're going to get one over Why squared? And when we do the same for our and I were gonna get negative two x over y cubed minus y. And so when we take the integral of our m i by d. X, we're going to get Native X over y squared. And this is because when we let's go, actually ah, well, yeah. Okay. So when we do the, um actually, no, this is going to be a positive and this we're going to get positive X over. Why squared? And so when we do the same for our, um And why over here, we're going to get a positive x over y squared and We're going to also track this by why squared over two. And so since this is since this is the same and the and this part is also the same. What we do is that we just write down the X over why square minus y squared over two.

To find the general solution. Let's first right this in differential operator for So that's gonna be be to the fourth minus 16 d Square plus 40 d minus 25. And then why is he got zero? So there. Now we confined the corresponding auxiliary equation. That's p of our is equal to our to the fourth minus 16 r squared plus 40 ar minus 25. So and we set that equal to zero. So to find the roots of this equation, I'm going to attempt to factor this first. So according to the rational root beer, um, all I need to do is look at the roots of 25 which are only 15 and 25 and we're gonna do the plus or minus of them. So I'm gonna write out, be, um, the coefficients. So I can use synthetic division to test the, um, to test the factors. Remember, Here, this is zero for the ark you term. Um, so there we go. Now, that's going to, um, Now I'm going to test a few factors, so I'm gonna try plus remind it's one first, so let's go with one. So I put in one, get 111 Then I add these two, which is one that I multiply. It's one that negative 15. So I get negative 15 that becomes 25. And then, uh, times 25 get zero. Since I have zero here, um, then that means zero is also a factor of or since I have zero here, that means one is a factor of this here. Now, here I have a cubic term. This is our cubed plus R squared minus 15 R plus 25. So I'm going to attempt to factor it some more. Okay, so I'm gonna factor this polynomial here now again by rational root beer. We only need to look at the factors to 25. So I'm gonna try plus or minus, um, one. So let's try one again. Let's see if one is a factor of this. A polynomial here again. So one bring down the one multiply one by one, we get one, then two and then so next we get one. Times two is to discuss negative 13 when they get a 13. Okay, so this doesn't This is not even zero. This is equal to 12 eso one is not going to be a factor, so let's try. Ah, different number. Let's try, uh, negative one. So negative one. Now. Time perspective one. That 00 negative. 15. 15. That's gonna be 40. So, again, not negative one. So now let's try five. Okay, So if we try five here we get five. This becomes six. Six times five is 30 and then negative. 15 times or plus 30. Gonna be negative. Are sorry. It would be positive. 15 and then 15 times five is 75. This is Ah, hundreds or not, Not five. Let's try. Negative five. So one times one is our Sorry one time saying, if I was thinking to five, this becomes negative for negative four. Uh, negative. Five times negative. Four is going to be positive. 20. This gives a positive five here. Positive five. Now, five times negative. Five is going to be negative. 25. Great. So now we found another factor of, um of this polynomial here. So then who we get are factored form? Um, it's gonna be r squared. Or so I are, uh, minus one for this first, this first factor here, then R plus five for this factor here. And then the remaining terms are gonna be the coefficients of R squared. Minus four are plus five is equal to zero. So let's try toe factor. This here. If we get the factors of five, um, they need to add up to five. But the only factors of five or one in five which clearly do not add up to four. So this cannot be factored. So we need to use a quadratic equation to solve this one here. So first, let's through the 1st 2 routes. The 1st 2 routes are 1st 2 routes are R equals one and negative five, and then here we're going to need to use court articulation. So negative b is for plus or minus and then square root of B squared, which is 16 minus four times A, which is one time, see is 20. So of 16 minus 20 here all over too. Okay, so this first part here gives us just to so is equal to two and then plus or minus Ah, 16 minus 20 is negative for square root of negative four is going to be equal to two I and then divided by two. Is this going to be plus or minus? I Okay, so then now these are our four factors. 15 and two, plus or minus. I. So our general solution, it's going to be why of X is equal to actually let me move this one and set up here. Okay? And then our general solution is going to be why of X is equal to and then we'll do the r equals once that c one e to the X, then plus C two e to the negative five x and then plus C three. And since we have a imaginary in solution, okay, so first we do, uh, this is a and then B is going to be one. We have C three e to the two X and then sign, uh, one x then plus C four be to the two x Times Co sign one X Hopes B X, but be is just one. So this is gonna be our general solution here.

So for this probably are asked Teoh solve for the differential equation. Our differential equation is d y over DX plus four y equals e to the negative three x So what I want to do first is I want to find my, um my integrating factor. So it's gonna be e and then to the power of the integral of in this case is going what's right in front of the why which is four. The X so the integral of four is going to be four x. So what I'm going to do is is multiply every term by each of the four x So eat of the four x d y over DX plus four eat of the four x Why equals E to the negative three x times eat of the four X So what I want to do is for the left hand side is this can condense down to D over DX and then times that integrating factor, which is eat of the four X times why and then this is going to equal e to the X. Okay, Now I'm ready to integrate some of put this in terms of DX on the right hand side. So now, because I have the DX and the integral those just cancel each other out. So I have eat of the four x y And then, of course, the integral e to the X is just eat of the X and then plus a constant. So now gonna divide by four X or eat of the four X. I'm sorry to all pieces someone I have X, um, minus for X. So it's gonna be e to the negative three x plus c over eat of the four X or you can rewrite. This is being Why equals e to the negative three x plus C to the even bigger forex whichever one, they're both the same thing.


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