I am trying to find a simple function $f(x, y)$ such that its derivative will have the form $$ f' = (1 - H) y dx + H x dy. $$ where $H$ is some constant.
The coefficients of the two terms make it hard to apply the product rule. I tried integrating by parts and the quotient rule, but i have not had much success.
Any suggestions or help will be much appreciated.
You know that $\frac{\partial f}{\partial x}= (1-H)y$. Integrating with respect to $dx$, you get $$ f = (1-H)xy+C(y)\tag 1 $$ Note that the constant which appears in integration can depend on $y$. Doing the same with the $y$ component, $$ f = Hxy + D(x)\tag2 $$ Equating these two, you can show that $H=1/2$ and $C(y)=D(x)=K$ for some constant $K$.
Edit: For a proof that the only possible $H$ is $H=1/2$, take equation (1) and compute $\frac{\partial^2}{\partial x\partial y}$ of both sides. You get that $\frac{\partial^2 f}{\partial x\partial y}=1-H$. Doing the same with equation (2), you get $\frac{\partial^2}{\partial x\partial y}=H$. This means $H=1-H$, so $H=1/2$.