
change of variable
I'm learning about the second order PDEs, and I've come across the transformation of variables I don't understand.
We have and a, b, c functions of class , which are not at the same time all equal to zero.
The equation in question is:
(*)
We introduce the change of variables:
,
How does the equation (*) now looks like?
I've tried googling change of variables, but I'm still lost.
Please, help.
Thank you.

By the chain rule,
Here, it is easy to see that and so .
Similarly,
You find the second derivatives by using those in succession:
.
and finally
Of course, since and , adding those equations, and so . Use that on the right hand side of the equation.