Link to actual problem [11612] \[ \boxed {y+x \left (x^{2}+y^{2}\right )^{2}+\left (y \left (x^{2}+y^{2}\right )^{2}-x \right ) y^{\prime }=0} \]
type detected by program
{"exactByInspection", "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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2}+y^{2}}{x^{4} y +2 x^{2} y^{3}+y^{5}-x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y^{4}}{4}+\frac {x^{2} y^{2}}{2}-\arctan \left (\frac {y}{x}\right )\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^{2}-y^{2}}{x^{4} y +2 x^{2} y^{3}+y^{5}-x} \\ \frac {dS}{dR} &= R^{3} \\ \end{align*}