Characteristic curves for speed one minus y squared
= Characteristic curves for speed one minus y squared
{title2=$dy/dx=\pm(1-y^2)$}
For $c(y)=1-y^2$ in the strip $|y|<1$, the travel-time coordinate is $s(y)=\operatorname{artanh}y$. The <characteristic curves> are
$$
\operatorname{artanh}y\pm x=\text{constant},
$$
equivalently $y=\tanh(C\mp x)$. Outside the strip the same separated <ordinary differential equation> gives <hyperbolic cotangent> branches, and the equilibrium curves $y=\pm1$ are also characteristic.