Can the PDE $u_{xx} + u_{xy} + u_{yy} = 0$ be separated by variables? 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.

You've almost answered your question already. If you divide by $XY$ you obtain:
$$ \frac{X''}{X}+ \frac{X'Y'}{XY}+ \frac{Y''}{Y}=0 $$
Hence,
$$ \frac{X''}{X}= \frac{X'Y'}{XY}- \frac{Y''}{Y} $$
Now, if it worked you'd be able to observe that each side of the expression depends only on $x$ (the l.h.s.) and only on $y$ (the r.h.s.). But, that fails here due to the mixed derivative term. Perhaps that is all is expected of you.
Maybe someone has a better answer? The hint about $45^o$ by paul garrett was encouraging you that this PDE is a quadratic form in $\partial_x, \partial_y$ and it can be diagonalized by changing to eigencoordinates. Personally, I would delighted if I had a student employ such a technique.
Note: $Q(x,y) = x^2+xy+y^2$ implies $A = \left[ \begin{array}{cc} 1 & 1/2 \\ 1/2 & 1 \end{array} \right]$ hence $det(A - \lambda I) = (\lambda -1)^2-1/4=0$. We find $\lambda_1 = 1/2$ and $\lambda_2 = 3/2$.This suggests you can choose coordinates $z,w$ such that the PDE takes the form $\frac{1}{2}u_{yy}+\frac{3}{2}u_{zz}=0$.
But, this is not allowed. Otherwise most classification questions in differential equations would be meaningless. Take the question of exactness for first order ODEs. You could always say it is exact by Pfaff's Theorem.
More to the point, rotation by 45 degrees (those are the eigencoordinates) is not a fair play in your game.
