Quasilinear Cauchy problem by characteristics

Let $u=u(x,y)$ be the solution to the Cauchy problem

$$(y+u)\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=x-y, x\in\mathbb R, y>0,\qquad u(x,1)=1+x, x\in\mathbb R.$$

Then which of the following statements is true?