$(x^2 + y^2 + x)dx - (2x^2 + 2y^2 - y)dy = 0$

Question

The question is $$(x^2 + y^2 + x)dx - (2x^2 + 2y^2 - y)dy = 0$$

I tries it by taking $\frac{dy}{dx}$ or by $t = x^2 + y^2$ but couldn't do further.

Answer 1

Re-write the equation as

$$(x^2+y^2)(dx-2dy)+(xdx+ydy)=0$$

and for $(x,y)\neq (0,0)$ dividing by $x^2+y^2$ gives

$$dx-2dy+\frac{xdx+ydy}{x^2+y^2}=0$$

Can you take it from here?