= Solution
Substitute the <partial derivatives> obtained above. The first-order terms cancel exactly:
$$
\begin{aligned}
u_{xx}-\tfrac12u_y-yu_{yy}
&=U_{\xi\xi}+2U_{\xi\eta}+U_{\eta\eta}
-\frac{U_\xi-U_\eta}{2\sqrt y}\\
&\quad-\left(U_{\xi\xi}-2U_{\xi\eta}+U_{\eta\eta}
-\frac{U_\xi-U_\eta}{2\sqrt y}\right)\\
&=4U_{\xi\eta}.
\end{aligned}
$$
The transformed <partial differential equation> is therefore
$$
\boxed{U_{\xi\eta}=0,\qquad \xi>\eta}.
$$
In these <characteristic coordinates>, the mixed <partial derivative> is the only surviving term.
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