Link to actual problem [7999] \[ \boxed {2 x^{2} y^{\prime \prime }+x \left (3+2 x \right ) y^{\prime }-\left (1-x \right ) y=0} \]
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}^{-x}}{x}\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}{4}, \frac {3}{4}, x\right ) {\mathrm e}^{-\frac {x}{2}}}{x^{{3}/{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{{3}/{4}} {\mathrm e}^{\frac {x}{2}} y}{\operatorname {WhittakerM}\left (-\frac {1}{4}, \frac {3}{4}, x\right )}\right ] \\ \end{align*}