1.6.1 Algorithm

Given Abel ode of first kind \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\), we first remove \(f_{2}\) as described above using transformation \(y=u\left ( x\right ) -\frac {f_{2}}{3f_{3}}\) which results in \(u^{\prime }\left ( x\right ) =k_{0}+k_{1}u+k_{3}u^{3}\). Now we check the Abel invariant \(\Delta \) defined as

\[ \Delta =-\frac {\left ( -f_{0}^{\prime }f_{3}+f_{0}f_{3}^{\prime }+3f_{0}f_{3}f_{1}\right ) ^{3}}{27f_{3}^{4}f_{0}^{5}}\]

And we assume the above is not constant. Only in this case we convert the ode to canonical form (if \(\Delta \) is constant, then it can be solved as shown above). So the goal now is to convert

\[ u^{\prime }\left ( x\right ) =k_{0}+k_{1}u+k_{3}u^{3}\]

to

\[ \eta ^{\prime }\left ( \xi \right ) =\Phi \left ( \xi \right ) +\eta ^{3}\left ( \xi \right ) \]

This is done as follows. We start by finding

\[ U=\exp \left ( \int k_{1}-\frac {k_{2}^{2}}{3k_{3}}dx\right ) \]

The result will be function of \(x\). i.e. \(U\left ( x\right ) \). Now we apply this transformation

\begin{align*} u\left ( x\right ) & =U\left ( x\right ) \eta \left ( \xi \right ) -\frac {k_{2}}{3k_{3}}\\ x & =\Phi \left ( \xi \right ) \end{align*}

To the ode \(u^{\prime }\left ( x\right ) =k_{0}+k_{1}u+k_{3}u^{3}\). This will result in

\[ \eta ^{\prime }\left ( \xi \right ) =\Phi \left ( \xi \right ) +\eta ^{3}\left ( \xi \right ) \]

Books say that \(\Phi \left ( \xi \right ) \) is defined parametrically where \(x\) is the parameter. Where

\begin{align*} \Phi \left ( \xi \right ) & =\frac {1}{f_{3}U^{3}}\left ( f_{0}-\frac {f_{1}f_{2}}{3f_{3}}+\frac {2f_{2}^{3}}{27f_{3}^{2}}-\frac {1}{3}\frac {d}{dx}\left ( \frac {f_{2}}{f_{3}}\right ) \right ) \\ \xi & =\int k_{3}U^{2}dx \end{align*}

Lets look at some examples showing how this is done.