Consider the diffusion equation uₜ = uₓₓ in { 0 < x < 1,0 < t < [infinity] } with u (0,t) = u ( 1, t) = 0 and u (x,0) = 4x(1-x).

Show that 0 < u (x, t) < 1 for all t > 0 and 0 < x < 1.