6.7 Separable first-order differential equations

Next we consider a type of differential equation called separable. The following equation appears to be an impossible case since we are mixing up $x$ and $y$ terms on the right:

But we can transfer the $y$ terms to the left-hand side to obtain ${\frac{1}{y^{2}}\frac{dy}{dx}=(x^{2}+1).}$ Now the left-hand side is just $\frac{d}{dx}\left({\frac{-1}{y}}\right)$, by the chain rule. Thus the equation has been converted to: ${\frac{d}{dx}\left(\frac{-1}{y}\right)=x^{2}+1.}$ This is an integrable equation! The general solution is given by integrating the left- and right-hand sides to obtain