I know that the method of characteristics is the technique to solve some PDEs. But I obtained an assignment to classify PDE (hyperbolic, elliptic or parabolic) using the very same method. I was wondering how to understand that the solution belongs to either an elliptic, parabolic or hyperbolic PDE.
For instance the transport equation has a linear characteristic curve. But how do I know that this implies that the PDE itself is hyperbolic.