Partial differential equation, mixed derivatives

Question

What can be concluded from following equation: $$\frac{\partial f(x,y)}{\partial x}-\frac{\partial g(x,y)}{\partial y} = 0$$ where $f(x,y)$ and $g(x,y)$ are functions of two independent variables $x,y$. Does it generally imply that $$\frac{\partial f(x,y)}{\partial x}=\frac{\partial g(x,y)}{\partial y}=a(x)b(y)$$ for some functions $a(x), b(y)$? Thanks for answer.

Answer 1

No.

It implies that there exists a function $F(x,y)$ such that $\partial_x F(x,y)=g(x,y),\partial_y F(x,y)=f(x,y)$. Since

$$\partial_x\partial_y F(x,y)=\partial_y g(x,y)=\partial_x f(x,y)=\partial_y\partial_x F(x,y)$$

In general you can not separate $F(x,y)$ into $a(x)b(y)$.