Link to actual problem [11921] \[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +8 y \left (x^{2}-1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 &= x \operatorname {BesselJ}\left (3, 2 \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselJ}\left (3, 2 \sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {BesselY}\left (3, 2 \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselY}\left (3, 2 \sqrt {2}\, x \right )}\right ] \\ \end{align*}