Should ρ, c, and k Be Included in the Heat Equation Solution?

[dirk_mec1]

Homework Statement

http://img3.imageshack.us/img3/5020/84513876dm0.png

The Attempt at a Solution

I found that $$f(t) =exp \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right)$$

Is this correct?

 

[dirk_mec1]

I thus only solved the ODE -problem.

 

[cristo]

Show your working, and it will be easier to check whether you are correct or not.

 

[dirk_mec1]

Show your working, and it will be easier to check whether you are correct or not.

OK, you got it:

$$\frac{ \partial{u} }{ \partial{t} } = \frac{df}{dt} \sin(\frac{m \pi x}{L})$$

$$\frac{ \partial{^2 u} }{ \partial{x^2} } = f(t) \left( \frac{-m^2 \pi ^2 }{L^2} \right) \sin(\frac{m \pi x}{L})$$
Therefore

$$f(t) =\mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right)$$Conclusion:
$$u(x,t)= u_0 + \mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) \cdot \sin(\frac{m \pi x}{L})$$

 

[Redbelly98]

Looks good, I'm just wondering one thing.

Should ρ, c, and k be in there somewhere, or do we assume units such that those are all =1?