Link to actual problem [6962] \[ \boxed {4 x^{2} y^{\prime \prime }+8 x \left (1+x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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}^{-x} \left (2 x \operatorname {BesselI}\left (1, x\right )+2 \operatorname {BesselI}\left (0, x\right ) x +\operatorname {BesselI}\left (0, x\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{x} y}{2 x \operatorname {BesselI}\left (1, x\right )+2 \operatorname {BesselI}\left (0, x\right ) x +\operatorname {BesselI}\left (0, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {{\mathrm e}^{-x} \left (2 x \operatorname {BesselK}\left (1, x\right )-2 \operatorname {BesselK}\left (0, x\right ) x -\operatorname {BesselK}\left (0, x\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} \sqrt {x}\, y}{-2 x \operatorname {BesselK}\left (1, x\right )+2 \operatorname {BesselK}\left (0, x\right ) x +\operatorname {BesselK}\left (0, x\right )}\right ] \\ \end{align*}