Link to actual problem [8116] \[ \boxed {2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y=0} \]
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 &= \sqrt {t}\, {\mathrm e}^{-\frac {t}{2}}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t}{2}} y}{\sqrt {t}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {WhittakerM}\left (\frac {1}{4}, \frac {1}{4}, \frac {t}{2}\right ) t^{\frac {1}{4}} {\mathrm e}^{-\frac {t}{4}}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t}{4}} y}{\operatorname {WhittakerM}\left (\frac {1}{4}, \frac {1}{4}, \frac {t}{2}\right ) t^{\frac {1}{4}}}\right ] \\ \end{align*}