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Using separation of variables method to solve the IBVP Ut=Uxxi 0<x<1 Itbp0 ulo,0)=0=ul1,0) u(x, 0) = 2 sin (Tx) 4(x,,0) = sin (2ux)62produces the general sol...

Question

Using separation of variables method to solve the IBVP Ut=Uxxi 0<x<1 Itbp0 ulo,0)=0=ul1,0) u(x, 0) = 2 sin (Tx) 4(x,,0) = sin (2ux)62produces the general solution ulx,0 = Zsin(nTx)[Ancos(nnt)+B,sin6ntt)]; n =1 Then the solution for the IBVP isOia ulx, t) = sin(Tx) cos(nt) + sin(2 Tx) sin(2nt) TU0 b. ulx,t) = sin(ux) cos(nt) + sin(2 Tx) sin(2nt) 2T 0 € ukx, 0) = 2 sin(Ix) cos(nt) + sin(2 Tx) sin(2tt) Old. ulx, t) = sin(Tx) sin(nt) + sin(2 Tx) cos(2nt)ulx; t) = 2sin(Tx) cos(nt) + sin(2 Tx

Using separation of variables method to solve the IBVP Ut=Uxxi 0<x<1 Itbp0 ulo,0)=0=ul1,0) u(x, 0) = 2 sin (Tx) 4(x,,0) = sin (2ux) 62 produces the general solution ulx,0 = Zsin(nTx)[Ancos(nnt)+B,sin6ntt)]; n =1 Then the solution for the IBVP is Oia ulx, t) = sin(Tx) cos(nt) + sin(2 Tx) sin(2nt) TU 0 b. ulx,t) = sin(ux) cos(nt) + sin(2 Tx) sin(2nt) 2T 0 € ukx, 0) = 2 sin(Ix) cos(nt) + sin(2 Tx) sin(2tt) Old. ulx, t) = sin(Tx) sin(nt) + sin(2 Tx) cos(2nt) ulx; t) = 2sin(Tx) cos(nt) + sin(2 Tx) sin(2nt) 2T ulx, t) = sin(tx) sin(It) + sin(2 Tx) cos(2nt)



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Use separation of variables to find the solution to the differential equation subject to the initial condition. $$\frac{d w}{d \theta}=\theta w^{2} \sin \theta^{2}, \quad w(0)=1$$

Hello. We have our number seven in this thing to use version of annual method. To solve this boundary value problem, finish about this initial value problems. Big bye bye. The X equal to cossacks in two areas to departed wife place Cynics before separating the recited as because it is the part way into it is the power cynics now separate the variable. It would be won by areas to the power Y. Dy equal to cossacks. Areas to the powers Synnex dx. We have integrated. Okay this is it is to power minus Y. Dy equal to if he All right. If you plug in sine X equal to T. So called 60 years will be able to DT our question. Our integration will turn out to be areas of power TDT from here it will be it is about minus well And into -1 will be equal two years about T plus C. And we have to find the value of C. For this we have been given the condition that when it's equal to zero, Y equal to zero. So if you plug in it will become minus one equal to of one place. Okay before that we had to write minus series of par -1 equal to it. Is the poverty T. Is cynics plus. Now we have to plan and execute where I could Okay -1 equal to one Plus C. So she will be equal to minus two. I'm looking over here minus. Used to par minus were equal to it is the power sin X minus two. Multiplying both sides by -1. It is the power minds well would be equal to two minus. It is the power syntax take Ellen both the sides equal to Ellen to minus eight days to the powers sign it simply. This would be minus Y equal to alan to minus it is the power syntax. So why will be equal to minus? Allen to minus areas to the power Cenex or one by Okay. And one by 2- areas to the power sign. Extra room visit dancer. And this will be true on living 2- areas to the power sign X Should be good than zero. So it is to the power cynic should be less than zero On less than two. If you take Ellen Synnex should be less than Ellen two. X should be less than sign verse. Alan two. Okay thank you.

So we have the difference like this. An artist, he is ableto signed treaty Homer signed treaty Homer t on any certainly is a narrow however to Z quartile. Who off by square before this competiton SDR writing t she wanted signed through tears. Same three capability. The artist book signed treaty Common Side Three people come up to your duty so interviewing voters Forget off the physical. Take yourself Sign JT duty on the Antichrist Sign 30 Duty hunting is DDT. This steak pudding find 3 30 degrees. Hey, tree close Prince Symmetric loss 30 Come home minus from Medicaid cause UT Comedy Square by two Yes, see, that'll be This is the syndromes artist. Now to find a particular Celeste with pertaining certainly isn't by our fiber to is it to horse play Scrabble? True? Yes. Quickly Manage number three cars 352 common minus and metric cause sleep I to I square by hate has see so c is equal to kind of work on the face square by two minus Classroom elevator is zero Jose Come on by square by eight this is the portal comma way Have he plays square by eight This is by my poor. There's a device paid substituting this in Munich was and we get off T is equal to what On December 3 cause treaty How minus from what he calls Haiti Coma Taste plus tau Whole color square by eight. This is a quantum Anderson battery, cause 18 still communist on that Because 80 this food, he's very, very toe this high square by it. This is the particular solution, Yeah.

We have on the six and this going to solve Do you lie by idea equal to causes? Quite Why? So this is one by because quite a wide you and because we're separating the variable. Now integrating This will be six square y do you a equal to dx? Then we will be equal to express C. Okay, so when X equal to zero, why call to zero? So let us play in here, zero Equal to zero plus C. so c equal to zero. Therefore the question would become then why equal to X? So why would be equal to 10 universe X? Which is true for any values of X? Thank you.

All right. So the problem here is why Prime Plus two T. Y. People's co sovereignty society over T. Square. And were given that um Why prime of uh sorry why of pi is equal to zero. All right. Get myself all sorted here. Okay. Um So first we can look at this and we see that is already in the form of Y. Prime plus P. F. T. Y giggles. GFT. Yeah. So it's gonna move on to the next step. And that is assigning PFT to be to over two. So P. A. T. We two or T. So we're gonna sign F. A. T. To be integral to poverty T. T. Which is to L. N. T. And so are multiplying factor is going to E. To the to L. N. T. It was just equal to T. Square. Sorry, I multiply that out into our equation. So we have T squared Y. Prime Plus two T. Y. illegal to co sign of T. Now it's a product rule on this side. Maybe you have T squared Y. Equal to co sign of T. Now the integral of D. Over DT We're going to integrate both sides of directed T. T. Square Y. D. T. It's the integral. Co sign the T. DT these cancel. And so it's gonna come out to being uh the integral of T squared Dy I'm just going to be able to see square. Why you go to sign of T. Plus C. All right. Why is he going to sign of T. Over if he squared plus C. Now he's dragging our initial position, which is why of pi is equal to zero. So we're gonna set this to zero equals sine of pi over pi squared plus C. Sign if I is zero. So this is going to be seeing 00. Really? They see. So why you'll see a sign of T. Two squared? Do there we go.


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