Re-writing higher order spatial derivatives as lower order system

I've been working with the following PDE:

$\displaystyle

\[

\nabla ^2 p = \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}

\]

$

What I would like to do is to re-write the second-order spatial derivatives in this PDE as first order derivatives.

This is what I have attempted, but I am uncertain as to whether this is correct. I introduce another variable $\displaystyle q$, and then:

$\displaystyle

\[

\frac{{\partial q}}{{\partial t}} = \nabla \cdot \vec p

\]

$

$\displaystyle

\[

\nabla \cdot \vec q = A\frac{{\partial p}}{{\partial t}} + Bp

\]

$

I reason that this is correct since

$\displaystyle

\[

\nabla \cdot \frac{{\partial q}}{{\partial t}} = \nabla \cdot \nabla \cdot \vec p = \nabla ^2 p

\]

$

$\displaystyle

\[

\frac{\partial }{{\partial t}}\left( {\nabla \cdot \vec q} \right) = \frac{\partial }{{\partial t}}\left( {A\frac{{\partial p}}{{\partial t}} + Bp} \right)

\]

$

Now is it reasonable to claim that the LHS of the two equations above are the same?