Canonical or standard form is the form for parabolic, hyperbolic and elliptic PDEs. There are as follows

parabolic

hyperbolic

modified hyperbolic

elliptic

where is lower order terms. Two go between the hyperbolic and modified hyperbolic introduce new coordinates .

Under the general change of variables

the first order derviatives transform as (the usual chain rule)

the second order derivatives transform as

If we substitute these three second order transforms into your PDE and target the modified form gives

Now, both are the same equation and both factor

Pick the first term for r and the second for s. These are first order PDEs and are easily solved giving

Now we'll pick easy

Under these change of variables you'll hit your modifed form. If you want regular form then choose

If you wish you can scale each by 2.