1.3.4.1.1 Algorithm Given \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\) let \(y=u\left ( x\right ) e^{\phi }\) where \(\phi =\int f_{1}dx\) and the ode becomes
Where \(F\left ( x\right ) =f_{0}e^{-\phi },G\left ( x\right ) =f_{2}e^{\phi }\). There is one special subcase here to consider. If \(F\left ( x\right ) \) turns out to be proportional to \(G\left ( x\right ) \) then the ode is separable. So this check should be done after completing the above. See second example below of one such example.