Link to actual problem [11308] \[ \boxed {x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 y \left (x +1\right )=x^{3}} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_method_1", "second_order_change_of_variable_on_y_method_2"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (x^{2}+2 y\right )}{2}\right ] \\ \end{align*}