TITLE: First Order Differential equation - separable form
QUESTION [6 upvotes]: I am trying to evaluate the equation:
$$y'=y\left(y^2-\frac12\right)$$
I multiplied the y over and tried to solve it in seperable form (M and N). The partial deritives did not work out to be equal to eachother so I am now stuck finding an integrating factor. Is this the right approach?

REPLY [2 votes]: This is also a Bernoulli equation, set $u=y^{-2}$ to get 
$$
u'=-2y^{-3}y'=-2+u\implies u=2+Ce^x(u_0-2)\implies y(x)=\frac{y_0}{\sqrt{2y_0^2+(1-2y_0^2)e^x}}
$$