= Solution
By the <superposition principle>, interchange $x$ and $y$ in the solution from part (c) and add the two solutions:
$$
\boxed{w(x,y)=u(x,y)+u(y,x)}.
$$
The first term supplies the required data on the positive $x$-axis and vanishes on the positive $y$-axis; the second does the reverse. Explicitly,
$$
\begin{aligned}
w(x,y)=\frac{4xy}{\pi}\bigg(&
\int_0^1\frac{v\,dv}{[(x-v)^2+y^2][(x+v)^2+y^2]}\\
&+\int_0^1\frac{v\,dv}{[(y-v)^2+x^2][(y+v)^2+x^2]}
\bigg).
\end{aligned}
$$
It is <harmonic> in the quarter-plane, has the stated boundary values, and decays in both unbounded directions.
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