Link to actual problem [9233] \[ \boxed {y^{\prime }-\frac {2 a \left (-y^{2}+4 a x -1\right )}{-y^{3}+4 a x y-y-2 y^{6} a +24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= y^{2}-4 x a, S \left (R \right ) &= \frac {y}{2 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 &= 0 \\ \eta &=\frac {256 a^{5} x^{3}-192 a^{4} x^{2} y^{2}+48 a^{3} x \,y^{4}-4 a^{2} y^{6}}{128 a^{4} x^{3}-96 a^{3} x^{2} y^{2}+24 a^{2} x \,y^{4}-2 a \,y^{6}+4 a x y -y^{3}-y} \\ \frac {dS}{dR} &= 0 \\ \end{align*}