= Solution
The <chain rule> for the new coordinates gives $\partial_x=\partial_\xi+\partial_\eta$ and $\partial_y=\partial_\xi-\partial_\eta$. Since the coordinates depend linearly on $x,y$, their coefficients are constant. Expanding the squares of these differential operators gives
$$
u_{xx}-u_{yy}=(u_{\xi\xi}+2u_{\xi\eta}+u_{\eta\eta})-(u_{\xi\xi}-2u_{\xi\eta}+u_{\eta\eta})=4u_{\xi\eta}.
$$
Hence the <partial differential equation> becomes $u_{\xi\eta}=1/4$. Integrating once in each variable introduces independent one-variable functions, giving the <general solution>
$$
\boxed{u=\frac{\xi\eta}{4}+F(\xi)+G(\eta)
=\frac{x^2-y^2}{4}+F(x+y)+G(x-y),}
$$
where $F,G$ are arbitrary twice-differentiable functions on the relevant characteristic intervals. Their sum includes an arbitrary additive constant.
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