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## Homework Statement

The temp. as a function of time of a metal rod obeys the following diff. eq.

[tex] \alpha^2 \frac{\partial^2u(x,t)}{\partial x^2} = \frac{\partial u(x,t)}{\partial t} [/tex]

Use separation of variables to find [tex] u(x,t) [/tex] in a rod of length 1 subject to the conditions [tex] u(0,t) = 0 , u(1,t) = 0 [/tex] and [tex] u(x,0) = 10 [/tex]

## Homework Equations

Seperation of variable I assume a solution of X(x)T(t) right?

I think the length of the rod is not needed?

## The Attempt at a Solution

Assume [tex] u(x,t) = X(x)T(t) [/tex]

Then I get [tex] \frac{\alpha^2}{X(x)}\frac{\partial^2 X(x)}{\partial x^2}= \frac{1}{T(t)}\frac{\partialT(t)}{\partialt} [/tex]

I then set them each equal to an arbitrary constant, -k^2 [tex]\frac{\alpha^2}{X(x)}\frac{\partial^2X(x)}{\parital x^2} = -k^2 [/tex] and [tex] \frac {1}{T(t)}\frac{\partial T(t)}{\partial t} = -k^2 [/tex]

I then solve each of them and get a function [tex] u(x,t) = (e^{-i\frac{k}{\alpha}x} +e^{i\frac{k}{\alpha}x})*(e^{-k^2t}-e^{k^2t}) [/tex]

Now I think thats correct, but Im not positive. I just cant seem to apply the first boundry condition to it....

Help!

PS, I thought this was more useful in the math than physics forum, but move it if necessary, thx

## Homework Statement

## Homework Equations

## The Attempt at a Solution

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