Link to actual problem [10092] \[ \boxed {x^{2} \left (y-1\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (y-1\right ) y^{\prime }-2 y \left (y-1\right )^{2}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= -\ln \left (x \right )+\operatorname {arctanh}\left (y\right ), S \left (R \right ) &= \frac {1}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {x^{2}}{2}, \underline {\hspace {1.25 ex}}\eta &= x \left (y -1\right )\right ] \\ \left [R &= x^{2} \left (y-1\right ), S \left (R \right ) &= \frac {2}{x}\right ] \\ \end{align*}