Solving a particulary tricky PDE

Question

I'm currently working on a problem that involves PDEs. I rarely work with them and have never formally learned any solution strategies other than the one I'm going to describe to you. However I somehow fail at the step of comparing the coeffictients:

Solving the PDE given by:

$$\partial_y u(x,y) = -x^2-y$$ and $$\partial_x u(x,y) = 2x - y$$

Therefore u(x,y) is given by:

$$u(x,y) =-x^2y - \frac{1}{2} y^2 + g(x)$$

and by:

$$u(x,y) = x^2 - yx + h(y)$$

Normally I'd just compare the results and see the answer, but currently I can't see anything :/

Answer 1

There is no smooth solution.By first equation, $$ \partial_x\partial_y u=-2x $$ By second, $$ \partial_y\partial_x u=-1 $$ They are not equal!