Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form k- a²u ax² hu = au at' 0 < x 0, ha constant. Find the temperature u(x, t) if the initial temperature is f(x) throughout and the ends x = 0 and x = L are insulated. See the figure. u(x, t) = = (± √ L 1 f(x) dx ) d x ) ( 0 )+(** L f(x) cos cos(x)dx) ηπ n=1 L Insulated 0° Insulated. )] L X 0° Heat transfer from lateral surface of the rod