Link to actual problem [6482] \[ \boxed {y^{\prime \prime }-\left (1+x \right ) y^{\prime }-y x=0} \] 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}^{-x} \operatorname {KummerM}\left (-1, \frac {1}{2}, \frac {1}{2} x^{2}+3 x +\frac {9}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\operatorname {KummerM}\left (-1, \frac {1}{2}, \frac {\left (x +3\right )^{2}}{2}\right )}\right ] \\ \end{align*}