I'm trying to solve a linear non-separable first order PDE of the form
$$ \nabla f(x,y,z) = g(x,y,z) \mathbf{v}, $$
where $g(x,y,z) = a_{11}x^2+a_{12}xy + \dots=\sum a_{ij}x_ix_j$ with $a_{ij}$ constant coefficients, and $\mathbf{v}$ is a constant vector.
I'm looking for any guidance on how to approach this.
At first, I tried taking the curl on both sides to get $\nabla g \times \mathbf{v}=0$ assuming $\partial_i \partial_j f = \partial_j \partial_i f$, but that implies $\mathbf{v}$ is parallel to $\nabla g$ for all values of $x$, $y$, and $z$, which seems inconsistent with $\mathbf{v}=\text{const.}$ It seems the solution (if any?) might take the form of nonseparable products on the variables, but all I could do so far was unsuccessfully proceed by trial and error.