 Solution

Integrate the canonical form of a second-order partial differential equation first in $\eta$ and then in $\xi$. The first integration gives $U_{\xi }=\eta +g^{\prime }(\xi )$, and the second givesThusThe 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.