$\left(y^2+x^2\right) d x-2 y d y=0$

$\left(y^2-x^2\right) d x-2 x y d y=0$

$\left(y^2-x^2\right) d x+2 y d y=0$

$\left(y^2+x^2\right) d x+2 y d y=0$

$y-x=c x^2$

$\tan ^{-1}\left(\frac{y}{x}\right)=\log \left(c x \sqrt{x^2+y^2}\right)$

$x+y=c x^2$

$\tan ^{-1}\left(\frac{y}{x}\right)=\log \left(c \sqrt{x^2+y^2}\right)$

$(\cos x+\sin x) y=\sin x+C$

$(\cos x+\sin x) y=\cos x+C$

$(1+\tan x) y=\cos x+C$

$\sec x(\cos x+\sin x) y=\sin x+C$

$\log \left|\frac{y^2-2 x^2}{x^2}\right|+\sqrt{2} \log \left|\frac{y-\sqrt{2} x}{y+\sqrt{2} x}\right| +2 \sqrt{2} \log |x|=C $

$\sqrt{2} \log \left|\frac{y^2-2 x^2}{x^2}\right|+\log \left|\frac{y-\sqrt{2} x}{y+\sqrt{2} x}\right| +2 \sqrt{2} \log |x|=C $

$\sqrt{2} \log \left|\frac{y^2+2 x^2}{x^2}\right|+\log \left|\frac{y+\sqrt{2} x}{y-\sqrt{2} x}\right| +2 \sqrt{2} \log |x|=C $

$\log \left|\frac{2 x^2-y^2}{x^2}\right|+\sqrt{2} \log \left|\frac{y+\sqrt{2} x}{y-\sqrt{2} x}\right| +\log |x|=C $