1.7.1 Algorithm

It is also possible to just work with Abel second kind all the time instead of working on solving the first kind. Given first kind Abel

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

This is converted to second kind using the transformation

\[ y=\frac {1}{u}\]

Which results in

\[ u^{\prime }u=k_{0}+k_{1}u+k_{2}u^{2}+k_{3}u^{3}\]

Where

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