Link to actual problem [8001] \[ \boxed {3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y=0} \]
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}{3}}}{x^{{1}/{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{{1}/{3}} {\mathrm e}^{\frac {x}{3}} 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}{6}, \frac {2}{3}, \frac {x}{3}\right ) {\mathrm e}^{-\frac {x}{6}}}{x^{{1}/{6}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{{1}/{6}} {\mathrm e}^{\frac {x}{6}} y}{\operatorname {WhittakerM}\left (-\frac {1}{6}, \frac {2}{3}, \frac {x}{3}\right )}\right ] \\ \end{align*}