Link to actual problem [12404] \[ \boxed {2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }-k y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not integer"}
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}} \sqrt {x}\, \operatorname {KummerM}\left (1+k , \frac {3}{2}, \frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\sqrt {x}\, \operatorname {KummerM}\left (1+k , \frac {3}{2}, \frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x}{2}} \sqrt {x}\, \operatorname {KummerU}\left (1+k , \frac {3}{2}, \frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\sqrt {x}\, \operatorname {KummerU}\left (1+k , \frac {3}{2}, \frac {x}{2}\right )}\right ] \\ \end{align*}