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2. Solve the differential First equation (IVP): (Te" + 3')da + Tzev dy check that the ODE is =0 exact y(2) = 0. Your final Then solve the IVP answer shou...

Question

2. Solve the differential First equation (IVP): (Te" + 3')da + Tzev dy check that the ODE is =0 exact y(2) = 0. Your final Then solve the IVP answer should be an implicitly delined curve; don't solve for y:

2. Solve the differential First equation (IVP): (Te" + 3')da + Tzev dy check that the ODE is =0 exact y(2) = 0. Your final Then solve the IVP answer should be an implicitly delined curve; don't solve for y:



Answers

For the differential equation $x y \frac{d y}{d x}=(x+2)(y+2)$, find the solution curve

Hello Rebel. You are going to solve the problem # 29. For first order linear differential equation. Paranal is equations. Okay so in the exact question we have to get the general solution for G. X plus two X minus Y squared. Do Y equal zero? Okay so let them okay let I am equal one and and equal two X minus Y squared since partial and partial Y equals zero partial um partial. Why equals zero and partial N. Partial X equal to. Okay so then okay parcel then partial X minus parcel. M. Okay sorry minus partial M. Partial. Why? Uh over um that's all equal to. Okay. Okay so let you X. Okay let you. Sorry you F. Y. Equals E. To the power two. The Y said that's what the E. Of two. Why? Okay then the equation will be E. R. Multiplying the equation. Boy that uh factor. So the question will be E. Well we can we write that? Sorry two More. Get more cloud. Education. It support to Y. Okay. Why? D. X plus E. Sorry mm So why of deploy Not to play two weeks multiply too, X minus Y squared. The Y Equal zero. Okay so let them one. Okay, so let M. one Equal E. It was about why? And N. One equal E. To the power to why multiply two x- Y swirled. Okay, so that's what will be equal to X. E. To the power to Y minus Y squared. Multiplied E. To the power to why? And find f. by integrating partial partially x equals m. one partially with respect to X. So holding why the constant then? So F. X. Y. Equal integration of E. To the power to Y. By the D. X. Sure equal uh X. Multiple I. E. To the power to Y plus B. Of Y. So P is a concept of integration that function of Why alone. So we're now find partially F. Partially why. Okay so partially partially why will be equal partially by partially why of X. Multiply E. To the power to Y plus B. Why? Okay that will be equal two X. E. To the power to Y plus partially be partially why? Okay so the function F satisfied partially F. Partially Y equals N. One can. Oh it's that partially us. Partially why equal and one. Okay so okay so sorry. Yes partial F. Partially why equal and one so T. Two. Sorry to X. E. To the power to Y plus partially be partially why equal two weeks multiple I. E. To the power to why minus Y squared eat with the power to why? Okay so partially be partially why will be equal negative wise words eat the power to Y. Okay so now we integrate dp by dy equal negative so when we uh integrate D. B. Bye. D. Y equal negative twice word E. To the power to why? With respect to why? Okay so by using the integration by parts twice so B. Of Y will be equal negative integration Y squared eat with the power to Y Dy well that will be call one negative 1/2 Y squared Plus one over y -1/4. E. Multiply E. To the power to Y plus C as a constant. So substituting for B. Y. In the function F of X. Y. That will be obtained. That F X Y will be equal X minus one Over two. Y squared Plus 1/2. Y -1/4. Multiply E. To the power to Y plus C. So the general solution for the differential equations is following that he is X -1/2. Wise words lost 1/2. Why? -1/4 plus C. E. To the bar negative two. Y equal zero. Okay, so that's a general solution for the given differential equation as E S C is constant. Okay. Thanks for watching and see you later.

In this problem. Give any question. As of the farm. MDX plus during the life since 20 just figure out the radio one night and and doing but the works minus doing my do it now. This is goods to work upon. Two X Y squared. That's why This is -1 -1 4, -1. This comes as -2. Whiteness. Sorry, this is -1 -2. 1 plus two X. Y. And this is why one plus to exercise this gets canceled with this. And we get this as -2 by by which is a function of Y alone. So the integrating factor is E to the part of minus to my life. The way it comes us E minus two. Lawn of why? It is one line. Why square? Well let's modify the integrating factor with the equation. To make it exact this is two X plus one. By why dx it's less who I am, minus x, Y, y. Square. Do you buy equals to zero. We need a function. Use such that Do you want the works equals two. The work's just one life. And do you but do I is equals two who I am minus? It's very nice. It's integrated this with respect to X. You get U. equals two X squared plus. Explain why plus the function of why? Now let's differentiate this with respect to why. And we get Do you buy the y equals two minus X. By Y squared plus F. Dash. Why comparing these two? We have I've touched by equals two. Why? What? F. Y. two. Y squared. Does the function you as equals 2? It's a square plus. Do Y plus Y squared is equal to a constant? You'll also get a constant. That key constant is incorporated. See itself? This is the solution not so.

Okay. So for this problem, I'm gonna go ahead and start off by trying to get everything into a dy dx format. What do I mean by that is I'm going to go ahead and move this term to the right hand side. So I'll have equals two X squared. Dy and then I can divide everything by X squared and D X. So I'll have Y squared divided by X squared plus. Why divided by X equals to do I defined by dx and right off the bat, I see an obvious substitution here. And that substitution is that U equals to Y divided by X. And do you do you X R. Sorry? Rewrite this before we go into that? All right. Why equals two you times X. Dy dx? It's kind of equal U plus X D U D X. All right. So let's go ahead and plug that in. So have you squared plus you equals two U plus x D u d x. These two cancel out right here by subtraction. And we're left with the U squared equals two X. D. U. D X. All right. So let me go ahead and move the X terms the left hand side and the U turn to the right hand side. So have one divided by X. D. X equals to one divided by you square. Do you? I'll have to integrate both sides. It's on the left hand side of left with will be left with natural log of X plus C. And this equals 2 -1 divided by you. And I can plug back in what I had for you. I said you is going to be Y divided by X. Natural log. The absolute value of X plus C equals two negative X divided by Y. All right. And we can go ahead and multiplied by a negative. So I'll have c minus natural log of X Equals two x divided by Y. And then we can also take the reciprocal, so we'll have y equals two X. Or whether I can divide X X divided by c minus natural log of X. So for this question, this will be your answer.

Hello everybody you are going to solve the problem number 37. For first order differential linear differential equations. Chapter of better knowledge equation. Okay, so we have to get the general solution for the following the french Regulation X squared plus two Y squared. Uh Okay. Dx by dy plus xy Equal zero. Okay so we can rewrite that function as following. X squared plus two Y squared dx. Okay. Plus X. Y. The Y equals zero. Okay, so both X squared plus two, Y squared and x Y. Are homogeneous of degree two. So let. Okay so let's why equal X. The then the Y equal X. The V plus the dx. Okay so substitute this end of the differential equation. Okay so the equation will be X squared Plus two x squared. His word. Okay the X plus X squared V. Multiply X D. V plus the D X equal zero. Okay. Okay so that will be X to the power two plus three, X squared B squared all D X Plus X. to the Power three. The D V equal zero. Okay, so By move simplification. So one over three the X plus the Over 1-plus 3 the squared D. V Equal zero. So by divide Bye X squared X. To the power 31 plus three V squared. Okay. Okay. Three D squared. Where next? Not equal zero. Okay, so so and making doing the integration. So bye making integration. So one over X dx integration of one over X dx plus Integration of the over 1-plus 3 disk word. The Okay that will be equal zero. Okay, so by continuing Lennox Plus 1/6, lend 1/1 plus three the squirt equal C. One. Okay so So 6 6 Lynn X plus len one or plus three three V squared equals six C. One. Okay so by more simplification Lynn X to the power six. Common factor. Multiply one plus three T squared Equal six c. 1. Okay so we can rewrite that. Mm. Okay sorry. Okay so we can get X to the power six multiply one plus three V squared Equals E. to the power six see one. So when we let that C Equal E to the Power six EC 1 C equals E. To the power six C. One. That Will be not equal zero. Okay so okay so We can rewrite that extra power six multiply one plus three Y squared over X squared equal. See. So it will be All we realized that X to the power six plus three X. To the power four Y. To the power tooth equal. See. Okay so it can be shown that x equals zero is a solution that X equals zero corresponds to see equals zero. So C can be any constant. Then judge the general solution is we can write that X to the power six Plus three X. to the power four Y squared equal. See when C is a constant. Okay so that's your solution. Thanks for watching and waiting you for any question about first order linear differential equations. So let's.


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