$\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ is non exact differential equation.
Here above equation satisfies $\partial M/\partial y=\partial N/\partial x$.
I know that above equation doesnot exist for $(0,0)$ but. How to prove that above given is not exact?
I don't know to how to show above?
Any Help will be appreciated.
First, $$ \frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy $$ is a differential form, not a differential equation. Now, if it is an exact differential form, then its integral along any closed curve must be zero. In particular, the integral of this form over a circle with radius $r$, centered at the origin, should be zero. Using polar coordinates $x=r\cos(\theta)$, $y=r\sin(\theta)$ and making the substitutions, you can see that the above differential form is equivalent to $d\theta$, but the integral of $d\theta$ over a circle is $2\pi$, not zero. Thus the differential form is not exact.