= Solution
{c}
Expanding the equation gives
$$
u_{tt}+2\rho u_{\rho t}+(\rho^2-1)u_{\rho\rho}+u_t+2\rho u_\rho=0.
$$
For the <hyperbolic partial differential equation> discriminant, $A=1$, $B=\rho$, $C=\rho^2-1$, so
$$
\boxed{B^2-AC=1>0\quad\text{everywhere}.}
$$
For a curve $\rho=\rho(t)$, the <characteristic hypersurface> equation is $(\rho')^2-2\rho\rho'+\rho^2-1=0$, or $\rho'=\rho\pm1$. Therefore the two families of <characteristic curves> are
$$
\boxed{\rho=-1+Ce^t,\qquad\rho=1+Ce^t,\qquad C\in\mathbb R.}
$$
The values $C=0$ include the characteristic lines $\rho=-1$ and $\rho=1$. Every curve in the first family approaches $\rho=-1$ as $t\to-\infty$; every curve in the second approaches $\rho=1$. The two curves through $(\rho,t)=(0,0)$ are $\rho=e^t-1$ and $\rho=1-e^t$, shown as the thick curves below. Equivalently, the <characteristic coordinates> are $e^{-t}(\rho+1)$ and $e^{-t}(\rho-1)$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-105.png]
{title=Characteristic families of the variable-coefficient wave equation}
{description=Blue and orange curves show the two families of <characteristic curves>, with arrows in the direction of increasing time. Dashed lines mark the characteristic lines $\rho=\pm1$. The thick curves pass through the origin. The initial segment and shaded forward strip indicate the region where the zero data force vanishing, as proved in the next part.}
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