Please bear with me - it has been years since I've touched PDEs and recently gotten back into the swing due to having started on a graduate program in physics.
Hints only
The general form for a second order linear PDE is given by
(1) $A\frac{\partial^2 \phi}{\partial x^2} + B\frac{\partial^2 \phi}{\partial x \partial y} + C\frac{\partial^2 \phi}{\partial y^2} + D\frac{\partial \phi}{\partial x} + E\frac{\partial \phi}{\partial y} + F\phi + G = 0$
where $A, B, C$ are function of $x, y$.
I am asked to classify the equation (whether elliptic, parabolic or hyperbolic)
(2) $\frac{\partial^2 \phi}{\partial z^2} + \frac{\partial \phi}{\partial z} + \frac{\partial^2 \phi}{\partial r^2} + \left ( \frac{1}{r} \right )\frac{\partial \phi}{\partial \theta} = 0$
Thinking aloud:
The equation in $(2)$ appears to be in cylindrical coordinates. So it seems the general course of strategy is to convert them to cartesian before making comparison with $(1)$.
Am I thinking in the right direction?