The following argument is from a [paper](http://arxiv.org/pdf/1305.3402.pdf) about the [Bendixson-Dulac Theorem](http://en.wikipedia.org/wiki/Bendixson%E2%80%93Dulac_theorem).



> 
> Consider a smooth differential equation on the plane
>  $$
> x'=g(x,y),\quad y'=h(x,y).
> $$
>  Suppose there exists a function $D(x,y)$ such that
>  $$
> (Dg)\_x+(Dh)\_y=0. 
> $$ 
>  *Then $D$ is an integrating factor and the system is integrable.* 
> 
> 
> 


A quick search for "integrable system" on Google returns results not satisfying. 


Could anyone explain what the last sentence in the argument above means?




---


In the [paper](http://arxiv.org/pdf/1305.3402.pdf) quoted above,


$$
g(x,y)=ax+bx^2+cxy,\quad h(x,y)=dy+exy+fy^2
$$


and $D(x,y)=x^ry^s$ for some $r,s$.



Integrating factors and integrability of an ODE system
$X=g\partial\_x + h\partial\_y$ is the vector field whose flow lines are wanted. 
  

$\omega=hdx - gdy$ is a 1-form with kernel the span of $X$. 
  

Also $D\omega$ has kernel the span of $X$ for any function $D$ which does not vanish anywhere. If $d(D\omega)=0$ (this is your condition) then $D\omega$ is a closed 1-form, thus exact on simply connected sets. So $D\omega = dF$ for a function $F$ which can easily be computed by line integrals.
Thus the wanted flow lines are contained in the level sets of $F$. 
  

Finally,
the time dependence of the flow has to be computed extra. 


