Differential EQNS: $y \, dx = \left[ x + (y^2 - x^2)^{1/2} \right] dy$

Please use proper groupings by using parenthesis. Hindi kasi malinaw ang problem.

The x don't have differential element, do you mean this?
$y \, dx = x + (y^2 - x^2)^{1/2} \, dy$

Or this?
$y \, dx = \left[x + (y^2 - x^2)^{1/2} \right] dy$

Solution
$y \, dx = \left[x + (y^2 - x^2)^{1/2} \right] dy$

$y(v\,dy + y\,dv) = \left[vy + (y^2 - v^2y^2)^{1/2} \right] dy$

$vy\,dy + y^2\,dv = \left[vy + y(1 - v^2)^{1/2} \right] dy$

$y^2\,dv = y(1 - v^2)^{1/2}\,dy$

$y\,dv = (1 - v^2)^{1/2}\,dy$

$\dfrac{dv}{(1 - v^2)^{1/2}} = \dfrac{dy}{y}$

$\displaystyle \int \dfrac{dv}{\sqrt{1 - v^2}} = \int \dfrac{dy}{y}$

$\arcsin v = \ln y + c$

$\arcsin \left( \dfrac{x}{y} \right) = \ln y + c$