Link to actual problem [9269] \[ \boxed {y^{\prime }-y^{2}-\frac {7 y x^{2}}{16}+\frac {y x}{2}-y^{3}-\frac {3 y^{2} x^{2}}{8}+\frac {3 y^{2} x}{4}-\frac {3 x^{4} y}{64}+\frac {3 y x^{3}}{16}=-\frac {1}{4} x +1+\frac {5}{128} x^{4}-\frac {5}{64} x^{3}+\frac {1}{16} x^{2}+\frac {1}{512} x^{6}-\frac {3}{256} x^{5}} \]
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 &= \frac {3}{4}+\frac {\left (x^{2}-2 x +8 y +8\right ) \left (x^{2}-2 x +8 y \right )^{2}}{512}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{\frac {3}{4}+\frac {x^{6}}{512}-\frac {3 x^{5}}{256}+\frac {\left (5+6 \textit {\_a} \right ) x^{4}}{128}+\frac {\left (-5-12 \textit {\_a} \right ) x^{3}}{64}+\frac {\left (6 \textit {\_a}^{2}+7 \textit {\_a} +1\right ) x^{2}}{16}+\frac {\left (-3 \textit {\_a}^{2}-2 \textit {\_a} \right ) x}{4}+\textit {\_a}^{3}+\textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}