I'm wondering if there is a function $f(x,y)$ such that
$\frac{\partial f}{\partial x}(x, y)=2x-3y$
and
$\frac{\partial f}{\partial y}(x, y)=3x+4y$
Integrating the first equation with respect to x and the second with respect to y, we get:
$\int (2x-3y)dx = x^2-3xy+c_1$ where $c_1$ can depend on y and:
$\int (3x+4y)dy = 2y^2+3xy+c_2$ where $c_2$ can depend on x.
At this point my argument is that in order to get a $-3y$ from a partial with respect to $x$, we need $-3xy$ in the function. But to get $3x$ from the partial with respect to $y$, we need $-3xy$. Therefore, in order to combine these 2 equations above, we would need $c_1$ to depend on $x$, which is not allowed, and $c_2$ to depend on $y$. How can I formalize this argument? Am I missing something? Do I have the right idea?