Link to actual problem [11294] \[ \boxed {4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y=0} \]
type detected by program
{"unknown"}
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}}{4}} \operatorname {WhittakerM}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {x}\, y}{\operatorname {WhittakerM}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {WhittakerW}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {x}\, y}{\operatorname {WhittakerW}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right )}\right ] \\ \end{align*}