Link to actual problem [10090] \[ \boxed {x \left (x +y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-y=0} \]
type detected by program
{"second_order_integrable_as_is"}
type detected by Maple
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_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*} \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (x +y\right )}{\left (\textit {\_y1} +2\right ) \textit {\_y1}}\right ] \\ \end{align*}