Hello, I have a question about example 4 from W2 L1:

We are given :$u_{t}+xu_{x}=xt $

after calculation we get:

$x=Ce^t$

$du=xt dt=Cte^t dt$, so $u=C(t-1)e^t+D=x(t-1)+D$,

but how did we get $D=\phi({xe^{-t}} )$? we know it is a constant, but why is D depended on $xe^{-t}$?

Also, why would the initial condition $u|_{t=0} =0 $ implies that $\phi({x}) =x$ ?

Thanks.