# What is the general solution of the differential equation dy/dx+y=xy^3?

What is the general solution of the differential equation $\frac{dy}{dx}+y=x{y}^{3}$?
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Derick Ortiz
Making the change of variable $y=\frac{1}{z}$ we have the new version
$\frac{dy}{dx}+y-x{y}^{3}=0\to \frac{x-{z}^{2}+zz\prime }{{z}^{3}}=0$ or
$x-{z}^{2}+zz\prime =0$
Now calling $\xi ={z}^{2}$ we have
$x-\xi +\frac{1}{2}\xi \prime =0$
Solving for $\xi$ we obtain easily
$\xi =\frac{1}{2}+x+C{e}^{2x}={z}^{2}$ then
$z=±\sqrt{\frac{1}{2}+x+C{e}^{2x}}=\frac{1}{y}$ then finally
$y=±\frac{1}{\sqrt{\frac{1}{2}+x+C{e}^{2x}}}$