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.