Let (x, t) satisfy the diffusion equation t = a 2xx for < x < , t > 0 with (x, 0) = f(x) in < x < . Show that the Laplace transform of (x, t) is given by (x, s) = 1 2a s Z f() exp s a |x | d, and hence deduce that (x, t) = 1 2a t Z f() exp (x ) 2 4a 2 t d. Hint: The lecture notes on the Fourier transform describe a Green's function for the ODE that should be useful