Link to actual problem [5563] \[ \boxed {x \left (x^{2}+1\right )^{2} y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference 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 \left (\frac {x +i}{-x +i}\right )^{\frac {1}{2}-\frac {\sqrt {1+i}}{2}} \operatorname {hypergeom}\left (\left [1+\frac {\sqrt {1-i}}{2}-\frac {\sqrt {1+i}}{2}, 1-\frac {\sqrt {1-i}}{2}-\frac {\sqrt {1+i}}{2}\right ], \left [1-\sqrt {1+i}\right ], \frac {x +i}{-x +i}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {x +i}{-x +i}\right )^{\frac {\sqrt {1+i}}{2}} y}{x \sqrt {\frac {x +i}{-x +i}}\, \operatorname {hypergeom}\left (\left [1+\frac {\sqrt {1-i}}{2}-\frac {\sqrt {1+i}}{2}, 1-\frac {\sqrt {1-i}}{2}-\frac {\sqrt {1+i}}{2}\right ], \left [1-\sqrt {1+i}\right ], \frac {x +i}{-x +i}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \left (\frac {x +i}{-x +i}\right )^{\frac {1}{2}+\frac {\sqrt {1+i}}{2}} \operatorname {hypergeom}\left (\left [1+\frac {\sqrt {1-i}}{2}+\frac {\sqrt {1+i}}{2}, 1-\frac {\sqrt {1-i}}{2}+\frac {\sqrt {1+i}}{2}\right ], \left [1+\sqrt {1+i}\right ], \frac {x +i}{-x +i}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {x +i}{-x +i}\right )^{-\frac {\sqrt {1+i}}{2}} y}{x \sqrt {\frac {x +i}{-x +i}}\, \operatorname {hypergeom}\left (\left [1+\frac {\sqrt {1-i}}{2}+\frac {\sqrt {1+i}}{2}, 1-\frac {\sqrt {1-i}}{2}+\frac {\sqrt {1+i}}{2}\right ], \left [1+\sqrt {1+i}\right ], \frac {x +i}{-x +i}\right )}\right ] \\ \end{align*}