= Solution
For the <method of characteristics>, parameterize by $x$; the <characteristic curves> satisfy
$$
\frac{dy}{dx}=-2x,\qquad \frac{du}{dx}=\cos x.
$$
Thus $x^2+y=C$ and $u-\sin x$ is constant on each <characteristic curve>. The general form in the quadrant is $u=\sin x+F(x^2+y)$. The <boundary condition> gives $F(s^2)=\cos s-\sin s$ for every $s\ge0$.
The <characteristic curve> through $(x,y)$ meets the boundary at $(s,0)$ with $s=\sqrt{x^2+y}\ge x$. Integrating backwards from that boundary value yields
$$
\boxed{u(x,y)=\sin x+\cos\sqrt{x^2+y}-\sin\sqrt{x^2+y}.}
$$
Indeed $\partial_x-2x\partial_y$ annihilates $x^2+y$ and sends $\sin x$ to $\cos x$, while on $y=0$ the expression reduces to $\cos x$. \b[Every characteristic in the quadrant meets the prescribed boundary, so its value is determined.] The displayed solution is smooth away from the origin. At the origin the boundary is characteristic and the solution is continuous but has a singular inward $y$ derivative; the differential equation is interpreted in the interior, with the boundary data attained continuously.
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