Differential Equations: $[x \csc (y/x) - y] dx + x \, dy = 0$

Solution (2)
$\left[ x \csc \left( \dfrac{y}{x} \right) - y \right] \, dx + x \, dy = 0$

$\left[ x \csc \left( \dfrac{vx}{x} \right) - vx \right] \, dx + x(v \, dx + x \, dv) = 0$

$(x \csc v - vx) \, dx + vx \, dx + x^2 \, dv = 0$

$x \csc v \, dx + x^2 \, dv = 0$

$\dfrac{dx}{x} + \dfrac{dv}{\csc v} = 0$

$\dfrac{dx}{x} + \sin v \, dv = 0$

$\ln x - \cos v = c$

$\ln x - \cos \left(\dfrac{y}{x} \right) = c$