The differential equation of the family of curves,

$x\left(y^{\prime}\right)^{2}=x+2 y y^{\prime}$

$x\left(y^{\prime}\right)^{2}=2 y y^{\prime}-x$

$x y^{\prime \prime}=y^{\prime}$

$x\left(y^{\prime}\right)^{2}=x-2 y y^{\prime}$