Given the 2nd order linear PDE
\begin{align} u_{tt}&=\frac{1}{9}u_{xx}, & &t>0,\ x \in \mathbb{R} \end{align}
Solution
We calculate the discriminant $$D=b^2-ac=\frac{1}{9}>0$$ So it is hyperbolic.
My question is how could I understand if I have to use $$\frac{dx}{dt} \quad or \quad \frac{dt}{dx}$$ to find the characteristics?