= Solution
Parametrize the <characteristics> by $s$, starting from $(x,y,u)=(1,r,r)$. Their <ordinary differential equations> are
$$
\frac{dx}{ds}=x,\qquad \frac{dy}{ds}=x+y,\qquad \frac{du}{ds}=1.
$$
They give $x=e^s$, $y=e^s(r+s)$ and $u=r+s$. Thus $s=\log x$ and $r=y/x-\log x$, so
$$
\boxed{u(x,y)=\frac yx\quad(x>0).}
$$
Indeed, $u_x=-y/x^2$, $u_y=1/x$ give $xu_x+(x+y)u_y=1$, 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 $0=1$.
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