ODE5.2

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

$y'+p(x)y=q(x)y^n\,$

And for these types of DE's a substitution of $z = y^{1-n}\,$ is recommended. In the present problem, $n=2\,$ and so

$z = y^{-1}\,$

$y = z^{-1}\,$

$y' = -\frac{z'}{z^2}\,$

Substituting these new variables into the original equation,

$-\frac{z'}{z^2} + \frac{x}{z} = \frac{x}{z^2}\,$

$z'-xz=-x\,$

Use the integrating factor $e^{\int -x dx} = e^{-x^2/2}\,$

$e^{-x^2/2}z' - e^{-x^2/2}xz = -e^{-x^2/2}x\,$

$\frac{d}{dx}[ e^{-x^2/2} z ] = -e^{-x^2/2}x\,$

$e^{-x^2/2} z = e^{-x^2/2} + c_1\,$

$z = 1+c_1e^{x^2/2}\,$

Undoing the substitution $z = y - 1$ ,

$y^{-1} = 1+c_1e^{x^2/2}\,$

$y = (1+c_1e^{x^2/2})^{-1}\,$