Problem:
Try to simplify $x^2\frac{\partial^2w}{\partial x^2}+y^2\frac{\partial^2w}{\partial y^2}+z^2\frac{\partial^2w}{\partial z^2}+yz\frac{\partial^2w}{\partial y\partial z}+zx\frac{\partial^2w}{\partial z\partial x}+xy\frac{\partial^2w}{\partial x\partial y}=0$ with transform $x=uv, y=vt, z=tu$, where $w$ is well-behaved enough.
Source: Григорий Михайлович Фихтенгольц
I found the method on the book (differentiating the equations in the transform to solve out $\partial w/\partial x$, etc) not easy to solve this problem, with a horrendous calculation. I wonder whether there's a sensible way to deal with the problem, without too pain and loss of rigor, for example, $\vartheta_s=s\dfrac\partial{\partial s}$?
Any help? Thanks!