Example 1 y′ = −xe−x − y + xe2xy3

\[ y^{\prime }=-xe^{-x}-y+xe^{2x}y^{3}\]

Comparing to

\[ y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\]

Shows that

\begin{align*} f_{0} & =-xe^{-x}\\ f_{1} & =-1\\ f_{2} & =0\\ f_{3} & =xe^{2x}\end{align*}

Applying transformation \(y=\frac {1}{u}\) gives

\begin{equation} u^{\prime }u=k_{0}+k_{1}u+k_{2}u^{2}+k_{3}u^{3} \tag {1}\end{equation}

Where

\begin{align*} k_{0} & =-f_{3}=-xe^{2x}\\ k_{1} & =-f_{2}=0\\ k_{2} & =-f_{1}=1\\ k_{3} & =-f_{0}=xe^{-x}\end{align*}

Hence (1) becomes

\[ u^{\prime }u=-xe^{2x}+u^{2}+xe^{-x}u^{3}\]

Which is second kind Abel