Link to actual problem [1308] \[ \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y=0} \] 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}{3}}}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{\frac {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^{\frac {1}{6}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{6}} {\mathrm e}^{\frac {x}{6}} y}{\operatorname {WhittakerM}\left (-\frac {1}{6}, \frac {2}{3}, \frac {x}{3}\right )}\right ] \\ \end{align*}