 Solution

Parametrize the characteristics by $s$, starting from $(x,y,u)=(1,r,r)$. Their ordinary differential equations areThey give $x=e^s$, $y=e^s(r+s)$ and $u=r+s$. Thus $s=\log x$ and $r=y/x-\log x$, soIndeed, $u_x=-y/x^2$, $u_y=1/x$ give $x{u}_x+(x+y)u_y=1$, and $u(1,y)=y$. The initial line is noncharacteristic, and its characteristics cover the half-plane $x>0$. The data do not determine a solution on $x<0$; a smooth solution through the origin is impossible because the differential equation there would read $0=1$.