Link to actual problem [6898] \[ \boxed {y^{\prime \prime }+2 y^{\prime } x +5 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
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 &= {\mathrm e}^{-\frac {x^{2}}{2}} \sqrt {x}\, \left (2 \operatorname {BesselI}\left (-\frac {3}{4}, -\frac {x^{2}}{2}\right ) x^{2}+2 \operatorname {BesselI}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right ) x^{2}-\operatorname {BesselI}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{\sqrt {x}\, \left (2 \operatorname {BesselI}\left (-\frac {3}{4}, -\frac {x^{2}}{2}\right ) x^{2}+2 \operatorname {BesselI}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right ) x^{2}-\operatorname {BesselI}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2}}{2}} \sqrt {x}\, \left (2 \operatorname {BesselK}\left (\frac {3}{4}, -\frac {x^{2}}{2}\right ) x^{2}-2 \operatorname {BesselK}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right ) x^{2}+\operatorname {BesselK}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{\sqrt {x}\, \left (2 \operatorname {BesselK}\left (\frac {3}{4}, -\frac {x^{2}}{2}\right ) x^{2}-2 \operatorname {BesselK}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right ) x^{2}+\operatorname {BesselK}\left (\frac {1}{4}, -\frac {x^{2}}{2}\right )\right )}\right ] \\ \end{align*}