Link to actual problem [9200] \[ \boxed {y^{\prime }-y^{2}-\frac {2 y x^{2}}{3}-y^{3}-y^{2} x^{2}-\frac {x^{4} y}{3}=-\frac {2}{3} x +1+\frac {1}{9} x^{4}+\frac {1}{27} x^{6}} \]
type detected by program
{"abelFirstKind", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Abel]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 1+\frac {\left (x^{2}+3 y +3\right ) \left (x^{2}+3 y \right )^{2}}{27}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{1+\frac {x^{6}}{27}+\frac {\left (3 \textit {\_a} +1\right ) x^{4}}{9}+\frac {\left (3 \textit {\_a}^{2}+2 \textit {\_a} \right ) x^{2}}{3}+\textit {\_a}^{3}+\textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}