Link to actual problem [6933] \[ \boxed {2 x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. 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 &= x^{\frac {3}{2}} \operatorname {KummerM}\left (1, \frac {7}{4}, \frac {x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {3}{2}} \operatorname {KummerM}\left (1, \frac {7}{4}, \frac {x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {3}{2}} \operatorname {KummerU}\left (1, \frac {7}{4}, \frac {x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {3}{2}} \operatorname {KummerU}\left (1, \frac {7}{4}, \frac {x^{2}}{2}\right )}\right ] \\ \end{align*}