Link to actual problem [5628] \[ \boxed {y^{\prime \prime }-y^{\prime }+y x^{2}=0} \] 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}{16}, \frac {3}{2}, i x^{2}\right ) {\mathrm e}^{\frac {1}{2} x -\frac {1}{2} i x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} {\mathrm e}^{\frac {i x^{2}}{2}} y}{\operatorname {KummerM}\left (\frac {3}{4}-\frac {i}{16}, \frac {3}{2}, i x^{2}\right ) x}\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}{16}, \frac {3}{2}, i x^{2}\right ) {\mathrm e}^{\frac {1}{2} x -\frac {1}{2} i x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} {\mathrm e}^{\frac {i x^{2}}{2}} y}{\operatorname {KummerU}\left (\frac {3}{4}-\frac {i}{16}, \frac {3}{2}, i x^{2}\right ) x}\right ] \\ \end{align*}