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A semi-infinite cylinder of metal lies along the positive x-axis with its sides and the face atinsulated. The initial temperature distribution is given by
where
and
are positive constants. Show that the temperature in the cylinder is given for
and
by
 = \frac{u_0}{2} e^{\frac{c^2 t}{L^2}} )
![\ \ \Biggl( \frac{x - \frac{2c^2t}{L}}{\sqrt{(4c^2t)}} \Biggr) \Biggr]](http://latex.codecogs.com/png.latex?\ \ \Biggl( \frac{x - \frac{2c^2t}{L}}{\sqrt{(4c^2t)}} \Biggr) \Biggr])
![\ \ \Biggl( \frac{x - \frac{2c^2t}{L}}{\sqrt{(4c^2t)}} \Biggr) \Biggr] \Biggr)](http://latex.codecogs.com/png.latex?\ \ \Biggl( \frac{x - \frac{2c^2t}{L}}{\sqrt{(4c^2t)}} \Biggr) \Biggr] \Biggr))
