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Gnen {() =Il where CJ <I <J {(c) periodic(-,#]Is f (x) eten odd function? f(r) Ecen fnd he Fourier Sro confdents coelliclentsIf it odd, Ind the2<I <-1 J...

Question

Gnen {() =Il where CJ <I <J {(c) periodic(-,#]Is f (x) eten odd function? f(r) Ecen fnd he Fourier Sro confdents coelliclentsIf it odd, Ind the2<I <-1 J ElaI<0 Given f (=) - #hich periodic 1-2,2 #ith period K0< I < 1 if 1 <I < Fourier Series coefficients for for n 1,2 and bn for n = 1,2 Using " the method euurulion variabls, solve the PDE ior ulI;Find theCjne - thc mcthod 4un consuniseparation of vuriablea LTE the PDE for u(z,g) for the secitic citeDying mnethod s

Gnen {() =Il where CJ <I <J {(c) periodic (-,#] Is f (x) eten odd function? f(r) Ecen fnd he Fourier Sro confdents coelliclents If it odd, Ind the 2<I <-1 J ElaI<0 Given f (=) - #hich periodic 1-2,2 #ith period K0< I < 1 if 1 <I < Fourier Series coefficients for for n 1,2 and bn for n = 1,2 Using " the method euurulion variabls, solve the PDE ior ulI; Find the Cjne - thc mcthod 4un consuni separation of vuriablea LTE the PDE for u(z,g) for the secitic cite Dying mnethod separation of wariables , salve the PDE for ulr,4) for the sPaif _ Cdtt when (e constant positive numhcr Usng the method scparation Wrihles Anlrr the PDE lor u(I,V): pcrod



Answers

Find the (real-valued) general solution to the differential equation. z??+8z?=0
z(t)=

So you have the phone division, Uh, you say that, uh, function f her morning more d region If, uh, didn't follow me through the civil route. You respect lex theft, remember? Why was the second creative off see of Africa? Respect Dizzy Izzy zero. Ah, but I know that these to greater can be seen us. It ain't virgins. The radiant ff Oh, she's been noted a small black squirt. So we see that harmonic isn't following happens satisfies their luck. Last equation that is called out to us equation. So these side is called couple of another man's aggression. So last equation equals you. So you have the following. If, um, the it is a bounded. Yeah. So what service, huh? Service? Yes. Then, uh, befalling happens. Degrading the greedy int in the unit. Nor more to s. So these were being the floods or the service out of the service. Cigna. Well, this is, uh this is equal to this year because, um bye. Divergence theorem. These were people toe giving over your homeboy the orgy, The big emergence off ground if, um better these diversions off, Brad. Easy goto they the blast crater. You should be zero. Sear them. So the volume or we'll be So it is his, uh, his violin. Better this. I'll be some numbers. It will be some Cranston. I'm zero. Which is the vault is here. So what are these doing? That brilliant faith out off the normal person out off this surface? Easy will go here on, uh, also you have be if, uh if you some are more nick morning function on Do come to me region. Yeah, well, uh, small surface, the service house of business. Uh, before what happens? Uh, then we have that are doing. We're this surface. Yeah, Know them soon, Grete. And two, this is the defense of surface is full doing, or the whole won't be ordered holding me. No, he's a squared. The right over the grabbed off the square. Also, these time here can be seen us in that pronto. If we took the break up, just know that No. So Well, what we gonna do here is to treat, decide, received. If perhaps we can. You can get it outside. So well, you have that. He's ah, job, Victor. So that victor that are within normal inside off. Oh, out off the surface over to surface. Bye. Diversions. Purim vehicle too. Review the virgins of that later. Well, the drug samples Yeah, multiply did really? So the equal toe doing that divergence over plan B and you'll be, but by the identities, identities or farces for the blind. Ah, uh, if a better view we have that piece. Did they bury himself? Some function that's a vector Fuel the origins of these people, though. Well, I have times on today rages well, first, plus, uh, granted Beth about product we, uh, or rectal feel so that, uh, well, we have these years. So you want to compute to die virgins? The virgin's off half right f is grad FSO selector. These three place of that serve these real people toe. You know, Jewel, we're be off f terms they virgins over the great interfere Plus, uh, no. The rating about bird with r f r f. He's great, in fact. So that's better. Devi will be able to I know that this part is already what you want these parties, huh? The greedy it off squared. So this part Morgan Day Savarin Angel Ortiz This is what we want and also note that these in every sense of grad fr that is precisely the definition off the separator. It's a very rare here is equal to the Lavery himself. But ah, we know that our kids are mine. Our money over f So being our money means that, uh, divergence. Oh, God, Yeah, you see Well, d'oh, do you so that these part of people do CEO of themself where volume well, for any continues functions here themselves were about invisible to zero. This time he's going so that these, well, people toe God. And that is that is precisely what I want that these term coincides with into a year over the morning. The oh, they know squared off Brilliant, yeah.

Okay. When a region, it's harmonic. We know devil possesion of that vector field, which is just like kind of ingredients square is equal zero. We're going to use that information to show that the surface in a role the greedy in of half that hat yes, is equal to zero. What we do here pretty simple. So by the divergence theorem, our breeding of EFTA and hat Yes, equal to triple integral of the over volume of space, of the divergence of whatever vectors in front that just happens to be the Grady int of, uh dp so pretty simple. The diversions of the grading is still a plush in. So a glossy is represented by this here. And we know that no harmonic region fellow Boston's equals zero. So we have internal of zero d. V. So that has to also be equal zero. And that's proof for our first part. For a second part, it is kind of the same thing. We just have to know that we have our initial function is just f grading of F dot and hat ds. Now, by our divergence, the room we know that we have divergence of our initial vector, which is, uh, raiding Beth TV. So we have to use kind of a changeable. For this part, we get triple volume in role of divergence. Excuse me? The greedy int Death times two Great in about now, according to product rule, we then have plus pef, sometimes a little bossy Annabeth. And now we know because the region is harmonic, that this term goes to zero. So we're just left with the integral of the grading of F squared value ingredients squared. Because in reality, these two are gonna be dotted into each other, which I raised a little bit here and then taps TV, which that's what we are trying to.

Are prone. 44 going to find on the outward heat flux. Given this and we have tea I Z z goto 100 plus X squared plus y squared Z squared and Caicos one. Alright, so f zero or a negative k Grady a t would you? Golden negative green tea which is the goto negative two x coming *** to Why come *** to Z? So then the grading of F is negative six. So six times the volume of the seer which is and D is ah sphere at the origin with a volume of for pie for three barbecues which is negative 24 pi over three, which is negative eight by

In this problem, a second dollar differential equation is given and we have to find general solution for this differential equation. So first thing that we can observe here that right inside of this differential equation is zero. Therefore this given differential equation is a homogeneous differential equation. So this is a homogeneous differential equation. Now recall the solution of for a homogeneous differential equation. We know that general solution is the complementary function of that solution. So we have to make an effort to find complementary function for this given homogeneous differential equation. No, to find complementary function. First we write auxiliary equation after given differential equation. So auxiliary equation will be now to write ancillary equations. We have to make a few substitution here. If we write uh this given different cell equation in uh in the form of double derivative or in form of dependent and independent variable. So this general death can be written as the square jed divided by here. We can observe that in the problem, it is mentioned, it is a function of time, therefore the independent variable is T. It means we have to differentiate it with respect to T. Therefore, first term general tests can be written as D squared divided by D t square plus eight gen. Death is representing first derivative of jet with respect to T. Therefore it will be deterred by DT as equals to zero, not right auxiliary equation. We substitute D square jet divided by D. T square with a capital D and digit by DT. Great. Sorry this question. But it is square with D. Square and digit by DT with D. So using this substitution, the auxiliary equation will be D Squire plus eight D. Is equals to zero. Now we have to find the root of this auxiliary equation. So root here that it is a very simple expression. So we can take common D. So D into deep plus eight will be zero. So this expression can be zero for two cases either D. Is zero or deep list 880 So if deep list 80 then in that case D. Z equals two minus eight. So here we got two routes of this equation, DZ equals to zero, and D. Is equals two minus it. No recall that how we write a complementary function. If we if if we know the rules of the auxiliary equation. So let rules of the auxiliary question is R. One and R two are route soft auxiliary equation. Then in that case complementary function as written as seven. He raised to the power are 20 plus C. Two. He raised to the power R two D. Read seven N. C. To R arbitrary constants. Yes. See an end C two. R. Our bit ready constants. Now here in this problem as we can see that roots are zero and minus eight. Therefore value of our win is zero and value of R two is minus eight. So we can put these values in the expression of complementary function. So our complementary function will be seven. He raised to the power zero into T. Less. See to be raised to the power minus eight into T. So this will reduce to seven. It is to the power zero. T. So so this term will be erased to the power zero and it is to the power zero is one. Therefore this will be simply seven into one that it's even plus C. Two days to the power minus 80. So this is the complementary function for the given auxiliary equation. Now in the starting of the problem, we had found that given differential equation watch homogeneous differential equation and solution of homogeneous differential equation is the complementary function itself. Therefore, general solution general solution uh That given differential equation will be jet as a function of T is equals to the complementary function that we have found, and it is C one plus C to be raised to the power minus 80 where C one and C two R arbitrary constants. So this is the solution for this problem.


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