First off, let's determine the type.

If

then

so it's hyperbolic. This means under a change of variable we should be able to transform you PDE to a standard form, i.e.

(I'm using r and s instead of

and

. lots - lower order terms). We now introduce first and secord order transforms. If

then

.

Substitute these into your PDE and group terms

.

In order to hit the target we need the coefficient of

and

to vanish. Thus,

,

or

They are the same so choose one for

and the other for

, i.e.

.

These are first oder PDEs, so by the method of characteristics, we fiind the solutions

which we choose the function

and

simple. So

(as you said).

Now put these into your transformations (1) and put all of these into you PDE and simply. See how that goes.