Link to actual problem [12269] \[ \boxed {\frac {x y^{\prime \prime }}{y+1}+\frac {y y^{\prime }-x {y^{\prime }}^{2}+y^{\prime }}{\left (y+1\right )^{2}}=\sin \left (x \right ) x} \]
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_xy]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)