Link to actual problem [1340] \[ \boxed {8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y=0} \] 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 {\operatorname {LegendreP}\left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}{\sqrt {2 x^{2}+1}\, x^{\frac {1}{8}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {2 x^{2}+1}\, x^{\frac {1}{8}} y}{\operatorname {LegendreP}\left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {LegendreQ}\left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}{\sqrt {2 x^{2}+1}\, x^{\frac {1}{8}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {2 x^{2}+1}\, x^{\frac {1}{8}} y}{\operatorname {LegendreQ}\left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}\right ] \\ \end{align*}