Say, I have a function $\phi$ of two variables ($x$ and $y$) My objective is not to solve this equation exactly but to look for the order of terms. I know that the scalar $\phi$ has following variation,
$\phi(x,y)= \phi(y) + \partial^2 _{x}\phi \ x^2$
Note that $\partial^2 \phi _{x}$ is only a function of y.
Now say my $f(y)$ is following,
$\Delta \phi = f(y)$
My objective here is to get some estimations of $f(y)$ based on the nature of $\phi$ I mentioned above. Is there anything I can say about $f(y)$ or/and $\partial_y f(y)$?