Link to actual problem [5621] \[ \boxed {y^{\prime \prime }+y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -2] \end {align*}
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}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= i \sqrt {\pi }\, \sqrt {2}-\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) \pi \,{\mathrm e}^{-\frac {x^{2}}{2}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{i {\mathrm e}^{\frac {x^{2}}{2}} \sqrt {2}\, \sqrt {\pi }-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x}\right ] \\ \end{align*}