Hi, I understand how to derive the solutions for Problem 3, but now I have a question. For example, in 3.a, the solution is

u=x/2 for 0<x<2t, but u(0,x)=x/2 which does not equal 0, the given initial condition. Also, du/dt(0,x)=0 which does not equal 1, the given initial condition. I know however that these initial conditions are satisfied for u=t as x>2t. Could you please explain this?