Question

Solve:

$xdx+ydy+\frac{xdy-ydx}{x^2+y^2}=0$.

Answer 1

Now,

$xdx+ydy+\frac{xdy-ydx}{x^2+y^2}=0$

or, $d\left(\frac{x^2+y^2}{2}\right)+\frac{\frac{xdy-ydx}{x^2}}{1+{\left(\frac{y}{x}\right)}^2}=0$

or, $d\left(\frac{x^2+y^2}{2}\right)+\frac{d\left(\frac{y}{x}\right)}{1+{\left(\frac{y}{x}\right)}^2}=0$

Integrating we have,

$\frac{1}{2}(x^2+y^2)+{\tan }^{-1}\left(\frac{y}{x}\right)=c$ [ Where $c$ is integrating constant]