Link to actual problem [5521] \[ \boxed {x^{2} y^{\prime \prime }+y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Irregular singular point"}
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 {BesselI}\left (\frac {i \sqrt {3}}{2}+1, -\frac {1}{2 x}\right ) {\mathrm e}^{\frac {1}{2 x}}}{\sqrt {x}}+\left (\left (1+i \sqrt {3}\right ) \sqrt {x}-\frac {1}{\sqrt {x}}\right ) {\mathrm e}^{\frac {1}{2 x}} \operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}, -\frac {1}{2 x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{2 x}} \sqrt {x}\, y}{-\operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}+1, -\frac {1}{2 x}\right )+\left (i \sqrt {3}\, x +x -1\right ) \operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}, -\frac {1}{2 x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {1}{2 x}} \operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}+1, -\frac {1}{2 x}\right )}{\sqrt {x}}+\left (\left (1+i \sqrt {3}\right ) \sqrt {x}-\frac {1}{\sqrt {x}}\right ) {\mathrm e}^{\frac {1}{2 x}} \operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}, -\frac {1}{2 x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{2 x}} \sqrt {x}\, y}{\left (i \sqrt {3}\, x +x -1\right ) \operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}, -\frac {1}{2 x}\right )+\operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}+1, -\frac {1}{2 x}\right )}\right ] \\ \end{align*}