Link to actual problem [11885] \[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (2 x^{2}+1\right ) y=0} \] 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 &= x \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {7}}{28}, \frac {3}{2}, \frac {i \sqrt {7}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {7}+1\right )}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} \left (i \sqrt {7}+1\right )}{4}} y}{x \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {7}}{28}, \frac {3}{2}, \frac {i \sqrt {7}\, x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {7}}{28}, \frac {3}{2}, \frac {i \sqrt {7}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {7}+1\right )}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} \left (i \sqrt {7}+1\right )}{4}} y}{x \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {7}}{28}, \frac {3}{2}, \frac {i \sqrt {7}\, x^{2}}{2}\right )}\right ] \\ \end{align*}