Solve the following Heat Equation: Show the details ofyour work: (30%) D.E. Ut(x,t) = c2uxx(x,t), 0 < x < ZTt,

If $u(x, t)=e^{a t} \sin (b x)$ satisfies the heat equation $u_{t}=$ $u_{x x},$ find the relationship between $a$ and $b$

Hello. In this problem we have two systems. The 1st 1 is a. With an initial temperature of T. Hey and another system B.

Also with temperature T. B. When both are in thermal contact with each other. So now we have a new system and the wretched equilibrium and the temperature of equilibrium S. T.

C. So if we are giving a piece of info about their magnitude of this temperature or how it compares to other temperature, we can say some piece of information regarding the initial temperatures of the two systems here. So first we have to keep in mind the fact that the temperature flows or the heat flows from that body with greater temperature. So heat flows from hot objects too cold objects for the higher temperature to the lower temperature. So here in case A We're being told that the temperature C.

Is less than the temperature eight. So what does this mean? The temperature that the system reached equilibrium and is less than the initial temperature of Aid. Which means that object has lost some heat or some energy. And uh it reached a lower temperature. And in that way we know that heat flow from A to B.

And this means that actually the temperature of object A. Was initially higher than the temperature of object B. Let's now look at the 2nd case. In the 2nd case, we are being told that the temperature C. Of the equilibrium is larger than the temperature of object A.

Which means here that object A increased in temperature meaning that it received some energy. And this energy definitely came from the contact it made with object B. So he flows from B to A. In that case, and this means that the temperature of object A. Was initially less than the temperature of object B.

So that's the answer, and that's our judgment in case A. And that's it in case B..

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So it was just very fighters and is correct here..