1) a) find the general solution u(x,t) to

ut+xux=u

b) what is the soln that satisfies the auxilliary conidtion u(x,0)=x^2?

2) suppose fx is a conitinuously differentiable vector field in R3 that satisfies the bound

|fx|<= 1/1+|x|^3

show that the triple integral (radiant).(Fdx)=0.

3) a solid, homogeneous object occupies the region D is contained R^3, and is completely insulated. its initial temperature in given by u(x,0)=f(x), for some function f. so u satisfies the heat equation ut=K().(delta u) with boundary condition partial of u/partial n=0 on partial of D. after a long time, the object reaches a steady, uniform temperature. in terms of f, what is this temperature?