Link to actual problem [2902] \[ \boxed {y^{\prime \prime }+y^{\prime } x +3 y=0} \] 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}} \left (-1+x \right ) \left (1+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{\left (-1+x \right ) \left (1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\sqrt {2}\, x^{2}-\sqrt {2}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {x^{2}}{2}} \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right )-2 x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{-2 x \,{\mathrm e}^{\frac {x^{2}}{2}}+\sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {\pi }\, \left (-1+x \right ) \left (1+x \right )}\right ] \\ \end{align*}