Link to actual problem [8021] \[ \boxed {9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y=0} \]
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 &= \frac {{\mathrm e}^{-\frac {x^{2}}{6}}}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{\frac {1}{3}} {\mathrm e}^{\frac {x^{2}}{6}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {WhittakerM}\left (\frac {1}{3}, \frac {1}{6}, \frac {x^{2}}{6}\right ) {\mathrm e}^{-\frac {x^{2}}{12}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{\frac {x^{2}}{12}} y}{\operatorname {WhittakerM}\left (\frac {1}{3}, \frac {1}{6}, \frac {x^{2}}{6}\right )}\right ] \\ \end{align*}