Link to actual problem [2416] \[ \boxed {x^{2} y^{\prime \prime }+x \left (-x^{2}+3\right ) y^{\prime }-3 y=0} \] 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 &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} \left (\operatorname {BesselI}\left (0, \frac {x^{2}}{4}\right ) x^{2}-\operatorname {BesselI}\left (1, \frac {x^{2}}{4}\right ) x^{2}-2 \operatorname {BesselI}\left (1, \frac {x^{2}}{4}\right )\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2}}{4}} x y}{\operatorname {BesselI}\left (0, \frac {x^{2}}{4}\right ) x^{2}-\operatorname {BesselI}\left (1, \frac {x^{2}}{4}\right ) x^{2}-2 \operatorname {BesselI}\left (1, \frac {x^{2}}{4}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} \left (\operatorname {BesselK}\left (0, -\frac {x^{2}}{4}\right ) x^{2}-\operatorname {BesselK}\left (1, -\frac {x^{2}}{4}\right ) x^{2}-2 \operatorname {BesselK}\left (1, -\frac {x^{2}}{4}\right )\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2}}{4}} x y}{\operatorname {BesselK}\left (0, -\frac {x^{2}}{4}\right ) x^{2}-\operatorname {BesselK}\left (1, -\frac {x^{2}}{4}\right ) x^{2}-2 \operatorname {BesselK}\left (1, -\frac {x^{2}}{4}\right )}\right ] \\ \end{align*}