Chapter 13.3, Problem 7P

A bar of length $l$ with insulated sides has its ends also insulated from time $t=0$ on. Initially the temperature is $u=x,$ where $x$ is the distance from one end. Determine the temperature distribution inside the bar at time t .Hints and comments: See the-discussion above and also Problem 2.14. Show that the $k=0$ solutions are $x$ and-constant (time independent). Note that here (unlike Problem 2.15) you do not need the extra solution (namely $x$) for $k=0$ since the final steady state is a constantand this is included in the solutions (3.10). Also note that we did need solutions in the discussion following (3.15) but were able to simplify the work by observing that these linear solutions simply give the final steady state.