Find the general integral of (x dx+y dy)/(√(x^2+y^2))=(y…

Find the general integral of $\dfrac{x\,dx+y\,dy}{\sqrt{x^2+y^2}}=\dfrac{y\,dx-x\,dy}{x^2}$.

Step 1: Recognize the left side

With $r=\sqrt{x^2+y^2}$: $dr=\dfrac{x\,dx+y\,dy}{\sqrt{x^2+y^2}}$, so the left side is exactly $d\big(\sqrt{x^2+y^2}\big)$.

Step 2: Recognize the right side

$d\!\left(\dfrac{y}{x}\right)=\dfrac{x\,dy-y\,dx}{x^2}=-\dfrac{y\,dx-x\,dy}{x^2}$, so the right side is exactly $-\,d(y/x)$.

Step 3: Combine

The equation becomes $d\big(\sqrt{x^2+y^2}\big)=-\,d(y/x)$, i.e.

\[d\!\left[\sqrt{x^2+y^2}+\frac{y}{x}\right]=0.\]