I just started reading Gerald Folland's book Fourier Analysis and Its Applications. I have a question about problem 1 from section 1.3.
The problem is the following.
- Derive pairs of ordinary differential equations from the following partial differential equations by separation of variables, or show that it is not possible.
$\quad$ (c) $\,$ $u_{xx} + u_{xy} + u_{yy} = 0$
Then if I try to separate variables by letting $u(x, y) = X(x)Y(y)$, I get the following equation
$ X''(x)Y(y) + X'(x)Y'(y) + X(x)Y''(y) = 0 $
For the other three equations corresponding to parts (a), (b) and (d) of the exercise I was able to separate the variables quite easily after doing the above substitution $u(x, y) = X(x)Y(y)$, but for this one I just don't think it's possible.
Question
Can the above equation be separated by variables? And if not, how would one prove that it can't be done?
I haven't been able to find examples where a particular PDE is shown not to be separable by variables.
Thanks for any help.