Consider the diffusion equation ut (x, t) = uxx(x, t) for t > 0 and r E R with initial condition u(x, 0) = (x). (1) Write down an explicit formula for the solution u(x, t) in the particular case or) = e-2x (2) For general initial condition o(x), show that lu( 2, t) 15 lo(y) |dy for all t > 0 and r E R. VAnt -00 (3) Suppose that the initial condition satisfies o(x) = '(x) for some function 4(x) satisfying 4(x) -+ 0 when x -+ co or x -+ -co. Show that ly(y) |dy for all t > 0 and r E R. Sen t J-00