Link to actual problem [7468] \[ \boxed {y^{\prime \prime }-y^{\prime } x^{2}+y x=x^{m +1}} \]
type detected by program
{"kovacic", "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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {\Gamma \left (\frac {2}{3}\right ) x^{3}-\left (-x^{3}\right )^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{\frac {x^{3}}{3}}-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) x^{3}}{\left (-x^{3}\right )^{\frac {2}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (-x^{3}\right )^{\frac {2}{3}} y}{-\left (-x^{3}\right )^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{\frac {x^{3}}{3}}+x^{3} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )}\right ] \\ \end{align*}