Link to actual problem [8871] \[ \boxed {x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 y^{2} x +x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries]]
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 &= x, \underline {\hspace {1.25 ex}}\eta &= \frac {5 y}{2}\right ] \\ \left [R &= \frac {y}{x^{\frac {5}{2}}}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}