by Comic Book Guy
Last Updated January 21, 2018 17:20 PM

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 $$ f = (1-H)xy+C(y) $$ Note that the constant which appears in integration can depend on $y$. Doing the same with the $y$ component, $$ f = Hxy + D(x) $$ Equating these two, you can show that $H=1/2$ and $C(y)=D(x)=K$ for some constant $K$.

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