ode internal name "abelFirstKind"
Currently the program detect this ODE and evaluates the Abel invariant only. 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 given by\[ \Delta =-\frac {\left ( -f_{0}^{\prime }f_{3}+f_{0}f_{3}^{\prime }+3f_{0}f_{3}f_{1}\right ) ^{3}}{27f_{3}^{4}f_{0}^{5}}\] In the case when \(f_{2}\neq 0\), then \(f_{2}\) is 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. \(\Delta \) can be constant (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.