Link to actual problem [10143] \[ \boxed {\left (2 y^{\prime } y^{2}+x^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{3}+3 y^{\prime } x +y=0} \]
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_poly_yn]]
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*}