Problem 02 | Separation of Variables

$2xy \, y' = 1 + y^2$

$2xy \dfrac{dy}{dx} = 1 + y^2$

$\dfrac{2y \, dy}{1 + y^2} = \dfrac{dx}{x}$

$\displaystyle \int \dfrac{2y \, dy}{1 + y^2} = \int \dfrac{dx}{x}$

$\ln (1 + y^2) = \ln x + \ln c$

$\ln (1 + y^2) = \ln cx$

$1 + y^2 = cx$
 

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