Link to actual problem [9214] \[ \boxed {y^{\prime }-\frac {-18 y x -6 x^{3}-18 x +27 y^{3}+27 y^{2} x^{2}+9 x^{4} y+x^{6}}{27 y+9 x^{2}+27}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]
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 {\left (x^{2}+3 y \right )^{3}}{9 x^{2}+27 y +27}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {9}{2 \left (x^{2}+3 y\right )^{2}}-\frac {3}{x^{2}+3 y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x^{2}+3 y \right ) \left (2 x^{5}+12 x^{3} y +18 x \,y^{2}+6 x^{2}+18 y +9\right )}{18 x^{2}+54 y +54}\right ] \\ \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=\frac {x^{6}+9 x^{4} y +27 x^{2} y^{2}+27 y^{3}}{6 x^{2}+18 y +18} \\ \frac {dS}{dR} &= {\frac {2}{3}} \\ \end{align*}