Can a Change of Variables Simplify This PDE to Normal Form?

[krocho]

Hi I have this problem
Reduce to normal form the following equation
x²d²f/dx²-y²d²f/dy²=xy.
First I guess by normal form , they mean a more standard equation, for example Laplacian.

I usually know what to do when the coefficients are constants, but in this case I don't know.I think it is possible to transform this into a Laplacian
, because we haven't seen the wave equation.
So I tried to do a change of variables, and to get the Laplacian I need to solve a system of four differential equations, do you think this is the way it should be done or there is an easier change of variables I should try.

thanks

 

[lippyka]

Yes, the change of variables approach is the correct way to go about solving this problem. In order to transform the given equation into its normal form, you will need to find a new set of variables that make the coefficients of the second-order derivatives constant. This can be done by solving a system of four differential equations.