Link to actual problem [13930] \[ \boxed {y^{\prime \prime }+y^{\prime } \left (x +2\right )+2 y=0} \] With the expansion point for the power series method at \(x = -2\).
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 {1}{2} x^{2}-2 x} \left (2+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{2 x} y}{2+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 2+i \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (2+x \right )}{2}\right ) \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{-\frac {1}{2} x^{2}-2 x -2} \left (2+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{2 x} y}{i {\mathrm e}^{-2} \sqrt {2}\, \left (2+x \right ) \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (2+x \right )}{2}\right )+2 \,{\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{2 x}}\right ] \\ \end{align*}