3.3.22 Abel first kind ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\)

ode internal name "abelFirstKind"

This ODE has the form

\begin{equation} y^{\prime }(x)=f_{0}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3} \tag {1}\end{equation}

Any of the following forms is called an Abel ode of first kind

\begin{align*} y^{\prime } & =f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\\ y^{\prime } & =f_{1}y+f_{2}y^{2}+f_{3}y^{3}\\ y^{\prime } & =f_{2}y^{2}+f_{3}y^{3}\\ y^{\prime } & =f_{0}+f_{2}y^{2}+f_{3}y^{3}\\ y^{\prime } & =f_{0}+f_{3}y^{3}\\ y^{\prime } & =f_{0}+f_{1}y+f_{3}y^{3}\\ y^{\prime } & =f_{2}y^{2}+f_{3}y^{3}\end{align*}

The case for both \(f_{0}(x)=0,f_{2}(x)=0\) is not allowed, else it becomes Bernoulli ode. Either \(f_{0}=0\) or \(f_{2}=0\) is allowed but not both at same time. The term \(f_{3}(x)\) must be there in all cases. When \(f_{2}=0\) then Abel invariant is defined as

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

In the case when \(f_{2}\neq 0\), then \(f_{2}\) is first removed from the original ode using the change of dependent variable \(y=u\left ( x\right ) -\frac {f_{2}}{3f_{3}}\). Now the new ode will not have \(f_{2}\) in it, and the above invariant can now be applied to it.

There are two possibilities when \(f_{2}=0\). Either \(\Delta \) can be constant (i.e. does not depend on \(x\)) or not constant (i.e. function of \(x\)). The constant invariant is the easier case and can be solved. The non constant case is not fully solved and only few cases can be solved analytically. This is not supported now.

If invariant \(\Delta \) is constant and \(f_{0}\neq 0\) (since we can not have both \(f_{0}=0,f_{2}=0\)) then the substitution

\[ y=\left ( \frac {f_{0}}{f_{3}}\right ) ^{\frac {1}{3}}u\left ( x\right ) \]

Results in a separable ode which can be solved. (See example below).

If \(f_{2}\) is not zero, then the first thing we do is transform the ode to remove \(f_{2}\). This is done using \(y=u\left ( x\right ) -\frac {f_{2}}{3f_{3}}\). What this means is that the new ode in \(u\left ( x\right ) \) will no longer have \(u^{2}\left ( x\right ) \) term in it. It will only have linear and \(u\left ( x\right ) ,u^{3}\left ( x\right ) \) in it only. Now we can apply the Abel invariant on this new ode.

After transformation to remove \(f_{2}\) we check the if Abel invariant  is constant or not. If not constant, then we check if it is Chini ode. I implemented solving Chini ode for special case only. Chini ode is similar to Abel but does not have the \(y^{2}\) term. This is why the transformation helps. This is the form of general Chini ode

\[ y^{\prime }=f_{0}\left ( x\right ) +f_{1}\left ( x\right ) y+f_{3}\left ( x\right ) y^{n}\]

When \(n=2\) then it is Riccati, and if \(n=3\) then it also Abel and for \(n>3\) it is general Chini. There is no general method to solve Chini for arbitrary \(n\). See my section on Chini ode on how to solve this ode for specific conditions.

References: Maple help pages.