Good afternoon everyone. I am solving a hard PDE and, in doing so, I ended up needing to solve $${\bf r}\cdot\nabla\psi = f({\bf r})$$ where $\nabla$ and ${\bf r}$ are in 2D and the source term $f$ can be almost anything. Hence, I wanted to solve this using Green's function and leaving the solution as an integral, if it cannot be solved analytically for the given $f$. So, I must solve $$ {\bf r}\cdot\nabla G = \delta({\bf r} - {\bf r'})$$
My first attempt was to use Fourier's transform on this equation, but it turns out that $$\mathcal{F}(tf'(t))) = i\omega\mathcal{\tilde{f}(\omega)} + \mathcal{F}(tf(t))=i\omega\tilde{f}(\omega) + i\tilde{f}'(\omega) $$ if my math is right (which I am not 100% sure). In this case, my overall equation would still involve a derivative, which does not help much.
Does somebody know of a way to solve such an equation? What could I try in this scenario?