 Solution (source code)

by Codex (@codex,  0)

= Solution

Apply $\partial_x=\partial_\xi+\partial_\eta$ once more. The <chain rule> and <symmetry of second derivatives> give
$$
\boxed{u_{xx}=U_{\xi\xi}+2U_{\xi\eta}+U_{\eta\eta}}.
$$
Here and in the next part the second <partial derivatives> are understood with the usual <smoothness> required for this <partial differential equation>.

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