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First; find the function N(x,Y) such that the following differential equation (2xZy+2eZxy?+2x)dx+N(x,y)dy=0is exact and N(O,y) =3y. Which of the following is the ge...

Question

First; find the function N(x,Y) such that the following differential equation (2xZy+2eZxy?+2x)dx+N(x,y)dy=0is exact and N(O,y) =3y. Which of the following is the general solution of the resulting exact differential equation?Yanitiniz:2xxtyextx+y=c+YeZx+x2+ ZecYex2+y=c2xYex2 + Y-cNone of the above

First; find the function N(x,Y) such that the following differential equation (2xZy+2eZxy?+2x)dx+N(x,y)dy=0 is exact and N(O,y) =3y. Which of the following is the general solution of the resulting exact differential equation? Yanitiniz: 2xxtyextx+y=c +YeZx+x2+ Zec Ye x2+y=c 2x Ye x2 + Y-c None of the above



Answers

Second-Order Differential Equations Find the general so-
lution to each of the following second-order differential equa-
tions by first finding $d y / d x$ and then finding $y$ . The general solu-
tion will have two unknown constants.
(a) $\frac{d^{2} y}{d x^{2}}=12 x+4$



(b)$\frac{d^{2} y}{d x^{2}}=e^{x}+\sin x$
(c) $\frac{d^{2} y}{d x^{2}}=x^{3}+x^{-3}$

So because of the fact that our differential equation involves a Y prime, the first step in solving this problem is going to be finding the first derivative of why So why prime is going to be equal to We're gonna have to do the chain well here. So, first we'll take the derivative of X squared over two and using the power rule, we know that this is going to be two x to the first power over to and this will be times E to the same exponents, um X squared over two. And so when we simplify, we find that why Prime is equal to X e the X squared over two. And so when we substitute of this into our differential equation, we know that Ah, why prime again is X e to the X squared over too. And this should be equal to X times. Why? And so we know that why is equal to e to the X squared over two. So when we plug all this into our differential equation, we get X E to the X squared over two is equal to x e to the x squared over to power. So since he's a reek willl that tells us that why is a solution to

Given the differential equation, I could get rid of that differential by using the integral. But you always want to separate your variables first. So I would need to multiply DX over to the other side. So I have d y equals y over x squared times DX Then if I want Teoh again separate all the variables I need to move this. Why? Because that why is still in the wrong side. So moving it over to the other side, I would have to divide it. So becomes one over why and then everything else tasing Now that I see, I only have wise on the left and I only have X is on the right. I'm ready to integrate on the left I have the form for the natural log member in a girl one over you is Ln of you so on that left I have Ln of why but be careful because you do not have Ln on the right side Here, watch this If I rewrite it, it's X to the native to that's not a natural log form. I can add one to this ex hone it which would make it extra the negative one and then divide by that new exponents, which to make a negative out front. So Ln of y equals negative, one extra, the negative one plus see some constant because I have an indefinite integral here. Now I get why by itself, if that's why I need to do, I would use E because he is the inverse of Ellen. And then this whole thing becomes the exponents on the other side. And maybe I ride it like native one over X is that X to the negative versus telling me it's actually on the denominator again? This is a good answer. Devonian. Maybe what my set up tells me to do. But I could rewrite it a little bit further to look honestly a little bit nicer based on some exponents rules here. So because these exponents are adding, we actually know that it had the same base that was multiplying. So it was e to the negative one over x times e to the C. That's my rules of exponents. This is helpful because I can replace this e to the C with just one big constant number, so I could actually write this final answer. Look, it just looks more simple. Fine. As why equals C e to the negative one over x

Number twelve Delilah Over the axe The cools lie over X square knot first separate over bus So where you want the wind? Why both sides? So you got a deal? I over one over y and multiple i d expos. Ay. So it's this party Don't have blond over X square. Look bloody X And you, integral beside his party. You've got leisure Lock off. Why? Absolute value, right? If this parted, get, um, this's axe to the inactive too. Why are X square so into broke that horse extras in anti juan to the ninety of on plus c So negative one of our explorers see so back to one over X pull us scene that's here for problem twelve. You can check by differentiate beside with respect to X So is part left part while we're wine like by t y o r d x equals lec tive x ninety one The security off that it was inactive. Valenti wants across expect you to and connected to two blood over at square see becomes era. Now if you must know why both sides got exactly this function. So that's the answer for number

We want to know which of the following functions are solutions City differentially freaking. So this is our front differential equation. So we want to find our wide double primed and our wives and put those in Syria. So let's start with a Tony too far and wide over prime. So first, let's remember that the derivative Ah, cause, uh is negative, son. And the derivative of Sign is positive co sign X. So just some odd interviews. Do you remember? So let's take artery the sign we're gonna get Coast Onyx and I protected a room that we get negative sign. It's so we're just gonna put those into our equation here so we'll have negative sign picks plus sign X people to sign X and more left and saw. This is zero, and it's a sign X These are not equal. Therefore, we know A is not our answer for bee. We're under the same thing to define what crime, which is negative. Signed X and then we're gonna get our wide double product, which would be bring out our negative review and sign is co sign so negative co sign it. Put those into our equation, will have negative co signed X Please Costa on X, you bring down our right side. Far left side of zero will know Right side is the sine X and these are not the same. Therefore be Zlata. Answer Joyce Fighter Chrissy, we need to find what crimes in one double prime placement or equation. So it's Yvonne. Why prime? We're going to use product will. So how was the first times that a review of the second close? The circuit times the derivative with first we said before that will have one have that. Oh sonic close one have signed it. We're gonna do the same thing to find my double prime. We have to use product rule for this portion here. So we say the first times that ever got a second plus the second times that during the first it's just one upside and they were students. Bring down this portion here, but it's derivative and not derivative is one have co sign it. So we simplify. This will have negative one happy ex tonics plus 1/2 co sign IX. That's one of those on exceptions one ex villages, but no one here or understood one. There So now we have We're why don't probably equation and r y equation and you just plug those into our function. We'll have negative 1/2 ex Sonics, close coast onyx lists or while equation and bring down our right hand side. So we simplify. We have a negative 1/2 X I next year on a positive here. So just counsel with each other and we have co sign X equals sine X over. This is not true. Therefore, seize letter answer toys either Does it look a Shame day? So even though we know that this has been an answer toys, we could still work it out. So we have our wives functions. I need to find our wives crime. So it's right there. You say the first times the ribbon of the second plus the spec It times that you're one of the first possible bodice will have positive 1/2 explore in it. Wilderness won her goes on it. Never use this upon our war double from equation. So take the first times that removed a second plus the second times that during your with the first and I was the derivative. This portion. Now it's like through this portion. So we have a minus one have sine X, but it changes. Sign because they removed could sign in the negative sign. But we simplify this. We'll have a one have ex co signed X and then just be one happens when half that's just one. So but signing? So now we have off the equation. So we need to plug into our differential equation here. Threw in a pregnancy. So you have want, huh? Thanks. Co sign it. Let's find it. Plus a negative one, huh? No. Co sign it and bring down our right hand side. Definitely. Look here and here. These subtract out they were left with signed X is able to sign it. Therefore, D is our answer choice.


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