I have a problem that says:
Let
$$\dot x = f(x,y) $$ $$\dot y = g(x,y) $$
be a 2-dimensional ODE with vector field $X:U\to \mathbb{R}^2 $, $X=(f,g)\in \mathcal C^1 (U)$, and $U$ an open star domain of $\mathbb{R}^2 $. X is integrable if there exists a $\mathcal C^1 $ function $H: U\to \mathbb R$ which is not constant over an open set, but it is constant over the solutions of the ODE.
- Prove that $X$ is integrable if, and only if, there exists a $\mathcal C^1 $ function $H$ such that
$$DH(x,y)X(x,y)=\frac{\partial H}{\partial x} (x,y) f(x,y)+\frac{\partial H}{\partial y} (x,y) g(x,y)=0 $$
If $\varphi(t)$ is a solution of the ODE, then $\dot \varphi = (f(\varphi),g(\varphi)) =X(\varphi)$, so if $X$ is integrable there exists a function $H$ such that $DH(\varphi)X(\varphi)=0$. Also, $DH(\varphi)X(x,y)=0, \; \forall x,y \in U$, but I don't know how to prove it in general. Maybe I should use that $U$ is a star domain, but I'm not sure because the problem has more sections.
Any help is welcome. Thanks!