How does one solve a "differential equation" for $\sigma$ of the form $ \sigma(v)w_i(v)={\partial \over \partial v_j}\left[\sigma(v)A_{ij}(v)\right] \quad i=1,\dots,n. $ where the summation convention applies.
$w,v$ are an $n$-D vectors, $\sigma$ is a scalar function, $A$ is an invertible $n\times n$ matrix?
Perhaps there is a general solution form? References (links) for the treatment of such an equation is also appreciated.
Thank you.
Added:
In light of drak's suggestion, here is a bit more
Some thoughts:
It might be friendlier to change "variables" to $A\sigma$?
Is there a more familiar expression for the index notation ${\partial \over \partial v_j}M_{ij}(v)$ such as one in terms of $\nabla$? It would seem to me that it is taking the divergence of each row of the matrix $M$.
Some more thoughts: since the function $\sigma$ appears on both sides of the equation, it is likely that it is an exponential.
A simplified version: What if we suppose that $A$ is a constant matrix?