= Solution
Integrate the <canonical form of a second-order partial differential equation> first in $\eta$ and then in $\xi$. The first integration gives $U_\xi=\eta+g'(\xi)$, and the second gives
$$
U(\xi,\eta)=\xi\eta+g(\xi)+h(\eta).
$$
Thus
$$
\boxed{u(x,y)=y^2-\ln^2|x|+g\bigl(y+\ln|x|\bigr)+h\bigl(y-\ln|x|\bigr).}
$$
The two arbitrary twice continuously differentiable <functions> describe the two <characteristic> families. Conversely, substituting this expression into $U_{\xi\eta}=1$ verifies every such solution. The formula applies on $x>0$ or $x<0$; independent choices can be made on the two half-planes. Smooth extension across the degenerate line $x=0$ requires extra compatibility and is not supplied by the logarithmic formula alone.
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