concept of canonical forms in PDEs - general question why the concept of deriving the canonical forms in PDEs is useful?
is it because it gives a way to solve some problems in a numerical way which otherwise would be difficult to solve using geometry? 

"The concept of deriving the canonical forms..." I think what is meant is, why is it useful manipulating a general 2nd order PDE into one of the canonical forms useful?  To answer my own question, because we can easily solve these canonical form PDE's analytically (at least on reasonable domains).  This then allows us to undo whatever transformations we made to get an analytical solution to the original PDE in question.
For example, it is easier solving $\nabla^2 u = F(x)$ for $x \in \mathbb{R^n}$ than it is solving the general elliptic PDE.
If the domain is complicated and one wishes to use a computer, numerical methods are more developed for efficiently solving the canonical equations as opposed to a general PDE.
