# Superficially simple vector differential equation problem

1. Feb 18, 2013

### cfp

Hi,

I have the following vector differential equation (numerator layout derivatives):

$$\frac{\partial e(v)}{\partial v}=\frac{1}{\beta} \frac{\partial w(v)}{\partial v} \Gamma^{-1}$$

where both $e(v)$ and $w(v)$ are scalar functions of the vector $v$, and where $\Gamma$ is a symmetric invertible matrix with all columns (and rows) summing to 1.

The naive solution would be $e(v)=\frac{1}{\beta} w(v) \Gamma^{-1}$, but this is incorrect since $e(v)$ is a scalar.

Clearly, when $\Gamma$ is the identity matrix, $e(v)=\frac{1}{\beta} w(v)$ is a valid solution. My question is, does a solution exist for any other value of $\Gamma$?

Tom