Link to actual problem [10047] \[ \boxed {y^{\prime \prime } \left (x -y\right )-\left (1+y^{\prime }\right ) \left ({y^{\prime }}^{2}+1\right )=0} \]
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*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {x -y}{x^{2}+y^{2}}, S \left (R \right ) &= \int _{}^{x}\frac {1}{\textit {\_a}^{2}-\frac {\left (x^{2}+y^{2}\right ) \left (1+\sqrt {-\frac {4 \left (x -y\right )^{2} \textit {\_a}^{2}}{\left (x^{2}+y^{2}\right )^{2}}+\frac {4 \textit {\_a} \left (x -y\right )}{x^{2}+y^{2}}+1}\right ) \textit {\_a}}{x -y}-\frac {\left (x^{2}+y^{2}\right )^{2} {\left (1+\sqrt {-\frac {4 \left (x -y\right )^{2} \textit {\_a}^{2}}{\left (x^{2}+y^{2}\right )^{2}}+\frac {4 \textit {\_a} \left (x -y\right )}{x^{2}+y^{2}}+1}\right )}^{2}}{4 \left (x -y\right )^{2}}}d \textit {\_a}\right ] \\ \end{align*}