Suppose that $f$ is a function of $x$ and $y$ such that (partial derivative in respect to $x$) $f_x = x+2y$ and (partial derivative in respect to $y$) $f_y = a + 3y$ where a is a constant. What does $a$ have to be, and why?
I am really confused as to how to approach this problem... any help would be much appreciated.
From the first one you have (integrating wrt $x$): $$f(x,y) = x^2 + 2xy + c(y)$$ and taking partial derivatives, $f_y = 2x + c'(y)$, but $f_y = a+3y$, so it must be $c'(y) = 3y$. That it says $a$ is a constant likely means constant wrt $y$, in which case $a=2x$ and it is not a problem. Then, integrating wrt $y$, $$f(x,y) = x^2 + 2xy + 3y^2/2 + K,$$ for some constant real $K$. However, if $a$ must be constant wrt $x$ and $y$ (e.g. $a_x = a_y = 0$), no solution exists...