Solving a simple system of partial differential equations in closed form

Question

I have the following system of partial differential equations -

$$\frac{\partial f(x,y)}{\partial x} = c_1(x^2+y^2) - c_2$$ $$\frac{\partial f(x,y)}{\partial y} = c_1((x-x_1)^2+(y-y_1)^2) - c_2$$

How do I go about solving these in closed form? If I try to naively integrate them, then the first one will involve $x^3$ while the second one will involve $y^3$.

Answer 1

Hint:

We get: $$\left\{\begin{array}{l}f(x,y)=c_1\frac{x^3}{3}+c_1y^2x-c_2x+g(y)\\ f(x,y)=c_1y(x-x_1)^2+c_1\frac{(y-y_1)^3}{3}-c_2y+h(x) \end{array}\right.$$

then $c_1=0$ (why?), and $g(y)=-c_2y$, and $h(x)=-c_2x$

If $c_1 \neq 0 $ then there is no solution.

If $c_1=0$ then $f(x,y)=-c_2(x+y)$