I have this PDE for the unknown functions $g(x,y)$ and $f(x,y)$ \begin{align} (\partial_x g)(\partial_y g) + (\partial_x f)(\partial_y f) = 0 \end{align} It is solved by $g=\arctan(y/x)$ and $f=\ln(x^2 + y^2)/2$. However I wanted to yield it by using Fourier Transform and am somehow stuck what the equations mean. So plugging in the Fourier-Transforms of g and f I get \begin{align} \int_{-\infty}^{\infty} {\rm d}k_x {\rm d}k_y {\rm d}k_x' {\rm d}k_y' \, \exp(ik_x x + ik_y y + ik_x' x + ik_y' y) \left(ik_x \tilde{g} \, ik_y' \tilde{g}' + ik_x \tilde{f} \, ik_y' \tilde{f}' \right) = 0 \end{align} Prime on the functions $\tilde{f}$ and $\tilde{g}$ is for primed argument. Now I integrate over the whole space of x and y to get 2 Delta functions. I'm then using $\tilde{f}(-k_x,-k_y)=\tilde{f}(k_x,k_y)$ because I assume the functions f and g be real. The result I get is $\tilde{g}^2 + \tilde{f}^2 = 0$, but what does this mean? I do not see how this yields the 2 solutions above?
Since I only have 1 equation there will be some freedom in the functions left, but still $\tilde{g}=\pm i \tilde{f}$ doesn't seem like a satisfying result, because then my functions are basically the same...