Consider an insulated uniform metal rod of length a with exposed ends and with thermal diffusivity 1. Suppose that at t = 0 the temperature profile is 1 0 (x,0) = 10 + sin 3x + 20 sin 5x = 2 sin 7x, but then the ends are held in ice at 0° C. When t is large, the temperature profile is closely approximated by a sinusoidal function of x whose amplitude is decaying to 0. What is the angular frequency of that sinusoidal function? (Hint: Start with the general solution to the heat equation with boundary conditions, and then match it to the given initial condition.)
