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[1O pts] Assuming product solution, ulx,t) = X(x) . T(t). use separation Of variables t0 derive LWO ODEs for X(x) and T(t). Do NOT solve the ODE $ just set up the e...

Question

[1O pts] Assuming product solution, ulx,t) = X(x) . T(t). use separation Of variables t0 derive LWO ODEs for X(x) and T(t). Do NOT solve the ODE $ just set up the equations0 u Dt

[1O pts] Assuming product solution, ulx,t) = X(x) . T(t). use separation Of variables t0 derive LWO ODEs for X(x) and T(t). Do NOT solve the ODE $ just set up the equations 0 u Dt



Answers

Find the solution of the differential equation that satisfies the given initial condition.
$ \frac {du}{dt} = \frac {2t + \sec^2t}{2u}, u(0) = 0 $

Okay, so we can get the generously shin bi integrations. That's the interval of E to the power T d T car the integral off the sign of t g t. So the 1st 1 is e to the power of tea plus C one common negative co sign of t plus two C two. So we know that at zero we have zero comma to zero results in zero comma to. So if we plug it in, we end up with C one is equal to native one and see to his equal to three. So this is your general solution than if you just plug in numbers. You end up with your final solution off t to the power of TV minus T Come on. Three T minus the sine of T minus three.

In the problem we have direction. Do you upon us choir that equals half integration in DT So it is integration you the par -2. Do you equal half tea? Bless Asi And this is -1 upon you equal. Do you want to place etc. So we we have UFT become -1 upon the upon to class C. Further you of Jiro equal minus one upon see so one equals minus one upon C or c equal minus one. Therefore you have to equal minus one upon The upon 2 -1. This is rigidness wanna bone 0.5 G minus one That equals one upon -0.5 T Plus one. So we have you FTS this and this is the answer.

On this question. We have a lot of a STD is equal. You did it Power, pinky. Finest. Come on, D squared. Minus one coma. Hold on. Then We have out of one is equal to 001 And we a bottle hard as one is available to. She was, you know, no hard Edward has taken Martinez. Do you Hardesty? About 80. He managed to. But these vermin has come upon Do your harder state is equally It is about a managed from a T square minus one month duty in getting with his hands. And here is a powerful theme and story T commenting recipes Kerman Ismael Beauty, Ridiculous beauty. This is a problem. Unviable goatee. Linus Caldwell de que a three minutes. Mhm como t. Let's see now we have any secondaries and orders. Hope Obama's secret toe Hamas zero comma zero. This is it for one. Why do we keep our see? Below normal one by three minus one Coma one has see. See, is a 00 minus one by two. Common minus. Good night, comma one. This is the portal three by two Cover story, right. It was by three over my last one our best, either. Very cordial one by community to team Anna's, too. On a teacher by three. My honesty, the That's straight by two. Come on, provide for my next one. This is a global Umberto muted power. Contain Honest to last People Committee Q by 3123 comity minus one A lot off Dealing with deep you know one by two to the car came and stood straight. I do particular humanist e that's 23 come up to you, minus one Beatty that says he called and take a send off my toe to team and stupid three by two. 80. How many degrees off if you buy three minus steep through by three. Duty commented. It is nothing on this one. Duty. This is a martyrdom when they hold about good team. Ministry has three by Pooky Coma Do pity party who by doing and it stays that way, too, has to write 80. Not a square cool minus. Let's say to this is our school. This is a village. No, we don't get our off on music. Wouldn't she? Was a little one. Andre, who will take our Children first. Revital comma one behaved very, Minister. That's true by three Come one very poor minus one Percy toe. So this is a good one. Bye. Food has two batteries. Uh huh. Yeah. 784 comb. Oh, that's one by four. Omar, man, it's on my to plus C two. It was safe to 0001 minus signed by 41 by four by two. This is a total when a seven by four among minus one day four comma you want to So they are a little p a substituting seem to get one day before you give out. T minus two. There's three very pootie coma duty off Overtly Highness, these vital has to buy three D commonly square by two. My nasty has on a seven by Holcomb Anderson by heart of a three by two This is a guru or off the When my fault it about a team and as toothless three very totally. Venice seven by four Comma his where we're too well and stay square way to this booty by three and has won by four committee square to dynasty has three way too. This is the particulars of this

Buying that this differential equation has this function as a solution. Let's take our derivatives to test that well, have you? Prime is equal to you. Let's see. See one into the tea. Estate is the same. That Iranian product world for the second. So well, I have seen too e to the T plus c Teoh t need to the team. Okay, We can group some coefficients together. We like the C to the t t in the team now we can take our second derivative. Then we'll just have one. Well, it is gonna be to see to see to the teat. Why is that? Well, when I do the product world for the next one and one of them, I'll leave the tea alone, take the dirt of either the t get itself, and the other one will take the derivative of tea. Yeah. Now, let's see if this satisfies the different Jill equation. I'll have one plus two C two, you know, T No, let's see, Plus into t unity. Then we're gonna subtract two times the first year. That so too. See one CTO eternity. Track C two to C two. See you to the team, and then we're gonna add the original function back and you didn t see to t It is a team. Okay, let's see what we get. Well, these two C two utilities, they're gonna cancel out with these negative to see two deputies. So let's get rid of what about this. See one into that he Well, there's two negative ceiling, you tease. So one of these will cancel out one of these. We'll have one remaining, but there's another scene. One either that he here. But somehow this cancels out. What else do we have there, too? Negative. C two t either The tees. Well, there's one here. There's one here. This is these two. Cancel out I get now. This one was cancelled out earlier, and we got to appreciate it out on rhythm.


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