Linear Matrix differential equation

Question

Let $A,B$ be non-singular matrices of dimension $n\times n$. Is there a way to solve the differential equation $$ f(x)Bx=A\nabla_x f(x)? $$ I've looked in many places and it doesn't seem to be obvious, likewise for the uni-variate case....

Answer 1

Define the variables $$\eqalign{ C &= A^{-1}B \cr \phi &= \log(f) \implies f = \exp(\phi)\cr }$$ Then $$\frac{\partial\phi}{\partial x} = \frac{1}{f}\frac{\partial f}{\partial x} = Cx$$ Now we can write $$\eqalign{ \int d\phi &= \int (Cx)^Tdx \cr }$$ If $C$ is symmetric, we can proceed to a solution $$\eqalign{ \phi &= \tfrac{1}{2}x^TCx + \phi_0 \cr }$$ If $C$ is not symmetric, then I am stuck.