Link to actual problem [6053] \[ \boxed {2 x^{2} y^{\prime \prime }+\left (x^{2}+5 x \right ) y^{\prime }+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not integer"}
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}{4}} \operatorname {WhittakerM}\left (\frac {5 i \sqrt {7}}{28}, \frac {5}{4}, \frac {i \sqrt {7}\, x}{2}\right )}{x^{\frac {5}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{4}} x^{\frac {5}{4}} y}{\operatorname {WhittakerM}\left (\frac {5 i \sqrt {7}}{28}, \frac {5}{4}, \frac {i \sqrt {7}\, x}{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}{4}} \operatorname {WhittakerW}\left (\frac {5 i \sqrt {7}}{28}, \frac {5}{4}, \frac {i \sqrt {7}\, x}{2}\right )}{x^{\frac {5}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{4}} x^{\frac {5}{4}} y}{\operatorname {WhittakerW}\left (\frac {5 i \sqrt {7}}{28}, \frac {5}{4}, \frac {i \sqrt {7}\, x}{2}\right )}\right ] \\ \end{align*}