Originally Posted by

**simplependulum** Does this question require you to apply the Bernouli's method ?

If not , I think we should use an other way : separating variables

$\displaystyle y' + xy = \frac{x}{y} $

$\displaystyle y' = \frac{dy}{dx} = x \left( \frac{1}{y} - y \right) $

by separating the variables ,

$\displaystyle \frac{y dy}{ 1-y^2 } = x dx $

integrating ,

$\displaystyle \frac{\ln( 1- y^2)}{2}= \frac{-x^2 + K }{2} $

$\displaystyle \ln(1 - y^2 ) =- x^2 + K $

$\displaystyle y = \sqrt{ 1 - e^{-x^2 + K }}$

Write $\displaystyle e^K = c $

$\displaystyle y = \sqrt{ 1 - ce^{-x^2}} $