I have a problem that I don't know how to solve it, It says:
Let $X=(f,g)\in \mathcal C^1(\mathbb{R})^2$ be a vector field and consider the system $\dot x = f(x,y), \; \dot y = g(x,y) $. If the system has an integrating factor $\mu(x,y) $, prove that $X$ has a first integral $H(x,y)$.
The hints and definitions that I have are:
$\mu(x,y) $ is a integrating factor iff $\mu(x,y)\neq 0, \; \forall (x,y)$ and $div (\mu X)=0 $
If $H$ is a first integral of $X$, then $DH(x,y)X(x,y)=0$
- If $divX=0$, then $X$ has a first integral (with some conditions of
the domain I think, but it works in this problem)
With all of this, we know that $\mu X$ has a first integral $\tilde H$, and I think I should prove that $\frac 1\mu \tilde H$ is the first integral of $X$ that I'm looking for.
Hence, I'm trying to prove that $D\left(\frac 1\mu \tilde H \right)(x,y)X(x,y)=0$, using that $D\tilde H(x,y) (\mu X)(x,y) =0$ and $div(\mu X)=0 $
On the one hand,
$$ div (\mu X)=\mu \;div X + \frac{\partial \mu}{\partial x} f+\frac{\partial \mu}{\partial y} g=0 $$
On the other hand,
$$D\left(\frac 1\mu \tilde H \right)(x,y)X(x,y)=-\frac1{\mu^2}\tilde H \left(\frac{\partial \mu}{\partial x} f+\frac{\partial \mu}{\partial y} g\right)+D\tilde H\cdot X = \frac 1\mu div X \cdot \tilde H$$
amd I'm not sure how I can continue, or if I'm in the correct way.
Any help is welcome! Thanks
