Question

Solve the equation:-
$\frac{xdx+ydy}{\sqrt{x^2+y^2}}=\frac{xdy-ydx}{x^2}$

Answer 1

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

as we know in differential equation

$d\left(\sqrt{x^2+y^2}\right)=\frac{xdx+ydy}{\sqrt{x^2+y^2}}$

& $\frac{xdy}{x^2}=d\left(\frac{y}{x}\right)$

so, $d\left(\sqrt{x^2+y^2}\right)=d\left(\frac{y}{x}\right)$

now integrate both side

$\int d\sqrt{x^2+y^2}=\int d\left(\frac{y}{x}\right)$

$\sqrt{x^2+y^2}=\frac{y}{x}+C$