Link to actual problem [1444] \[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+8\right ) y^{\prime }+\left (3 x^{2}+5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference 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}}{8}} \left (\operatorname {BesselI}\left (0, \frac {x^{2}}{8}\right )+\operatorname {BesselI}\left (1, \frac {x^{2}}{8}\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{8}} \sqrt {x}\, y}{\operatorname {BesselI}\left (0, \frac {x^{2}}{8}\right )+\operatorname {BesselI}\left (1, \frac {x^{2}}{8}\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}}{8}} \left (\operatorname {BesselK}\left (0, \frac {x^{2}}{8}\right )-\operatorname {BesselK}\left (1, \frac {x^{2}}{8}\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{8}} \sqrt {x}\, y}{\operatorname {BesselK}\left (0, \frac {x^{2}}{8}\right )-\operatorname {BesselK}\left (1, \frac {x^{2}}{8}\right )}\right ] \\ \end{align*}