2) Consider a semi-infinite pipe of stationary water. Dye is injected into the pipe at the end, and diffuses throughout the water. At each time &, dye is injected into the water with constant flux 1. The total density of dye is given by u (c, (), which solves 1 (z,t) - War (I, t) = 0, t,2>0, (x, 0) = 0,2>0 us (0, 1) - 1, t30. Solve for u(x, () for all t, > > () in terms of Erf(x) if necessary. Does u (x, () converge as t -+ oo? Plot your solution for t = 0. 1. 10