$2xy \, y' = 1 + y^2$
$2xy \dfrac{dy}{dx} = 1 + y^2$
$\dfrac{2y \, dy}{1 + y^2} = \dfrac{dx}{x}$
$\displaystyle \int \dfrac{2y \, dy}{1 + y^2} = \int \dfrac{dx}{x}$
$\ln (1 + y^2) = \ln x + \ln c$
$\ln (1 + y^2) = \ln cx$
$1 + y^2 = cx$
when $x = 2$, $y = 3$
$1 + 3^2 = 2c$
$c = 5$
then,
$1 + y^2 = 5x$
$y^2 = 5x - 1$
$y = \sqrt{5x - 1}$ answer