Link to actual problem [9276] \[ \boxed {y^{\prime }-\frac {-128 x y-24 x^{3}+32 x^{2}-128 x +512 y^{3}+192 y^{2} x^{2}-384 y^{2} x +24 y x^{4}-96 x^{3} y+96 x^{2} y+x^{6}-6 x^{5}+12 x^{4}}{512 y+64 x^{2}-128 x +512}=0} \]
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 {x^{6}-6 x^{5}+24 x^{4} y +12 x^{4}-96 x^{3} y +192 x^{2} y^{2}-8 x^{3}+96 x^{2} y -384 x \,y^{2}+512 y^{3}-16 x^{2}+32 x -128 y -128}{64 x^{2}-128 x +512 y +512}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {64 x^{2}+512 \textit {\_a} -128 x +512}{x^{6}-6 x^{5}+\left (24 \textit {\_a} +12\right ) x^{4}+\left (-96 \textit {\_a} -8\right ) x^{3}+\left (192 \textit {\_a}^{2}+96 \textit {\_a} -16\right ) x^{2}+\left (-384 \textit {\_a}^{2}+32\right ) x +512 \textit {\_a}^{3}-128 \textit {\_a} -128}d \textit {\_a}\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 &= 1 \\ \eta &=-\frac {x}{4}+\frac {1}{4} \\ \frac {dS}{dR} &= \frac {4 R +4}{4 R^{3}-R -1} \\ \end{align*}