Link to actual problem [10107] \[ \boxed {\left (y^{2}+x^{2}\right ) y^{\prime \prime }-2 \left ({y^{\prime }}^{2}+1\right ) \left (y^{\prime } x -y\right )=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{y}\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {x^{2}+y^{2}}{x}, S \left (R \right ) &= -\frac {y^{2}}{\left (x^{2}+y^{2}\right ) \sqrt {y^{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {y^{2}}{2}+\frac {x^{2}}{2}, \underline {\hspace {1.25 ex}}\eta &= x y\right ] \\ \left [R &= \frac {y}{x^{2}+y^{2}}, S \left (R \right ) &= \frac {2 \left (\frac {y^{2}}{x^{2}+y^{2}}-1\right ) y}{\left (x^{2}+y^{2}\right ) \sqrt {-\frac {y^{2} \left (\frac {y^{2}}{x^{2}+y^{2}}-1\right )}{x^{2}+y^{2}}}}\right ] \\ \end{align*}