Link to actual problem [11884] \[ \boxed {y^{\prime \prime }+8 y^{\prime } x -4 y=0} \] 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}^{-2 x^{2}} x^{\frac {3}{2}} \left (\operatorname {BesselI}\left (-\frac {1}{4}, 2 x^{2}\right )+\operatorname {BesselI}\left (\frac {3}{4}, 2 x^{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 x^{2}} y}{x^{\frac {3}{2}} \left (\operatorname {BesselI}\left (-\frac {1}{4}, 2 x^{2}\right )+\operatorname {BesselI}\left (\frac {3}{4}, 2 x^{2}\right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-2 x^{2}} x^{\frac {3}{2}} \left (\operatorname {BesselK}\left (\frac {3}{4}, 2 x^{2}\right )-\operatorname {BesselK}\left (\frac {1}{4}, 2 x^{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 x^{2}} y}{x^{\frac {3}{2}} \left (\operatorname {BesselK}\left (\frac {3}{4}, 2 x^{2}\right )-\operatorname {BesselK}\left (\frac {1}{4}, 2 x^{2}\right )\right )}\right ] \\ \end{align*}