Link to actual problem [6429] \[ \boxed {y^{\prime \prime }+2 y^{\prime } x -y=x} \] 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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2}}{2}} x^{\frac {3}{2}} \left (\operatorname {BesselI}\left (-\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselI}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x^{\frac {3}{2}} \left (\operatorname {BesselI}\left (-\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselI}\left (\frac {3}{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}} x^{\frac {3}{2}} \left (\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )-\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}{x^{\frac {3}{2}} \left (-\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}