Link to actual problem [9390] \[ \boxed {y^{\prime \prime }-x^{2} \left (x +1\right ) y^{\prime }+x \left (x^{4}-2\right ) y=0} \]
type detected by program
{"kovacic"}
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{3}}{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {x^{3}}{3}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{3}}{3}} \left (\int {\mathrm e}^{\frac {x^{3} \left (3 x -4\right )}{12}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{3}}{3}} y}{\int {\mathrm e}^{\frac {1}{4} x^{4}-\frac {1}{3} x^{3}}d x}\right ] \\ \end{align*}