Link to actual problem [14803] \[ \boxed {\left (x^{2}-3 x -4\right ) y^{\prime \prime }-\left (x +1\right ) y^{\prime }+y \left (x^{2}-1\right )=0} \] 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}^{-i x} \operatorname {KummerM}\left (\frac {3}{2}+\frac {3 i}{2}, 3, 2 i x -8 i\right ) \left (x -4\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{i x} y}{\left (x -4\right )^{2} \operatorname {KummerM}\left (\frac {3}{2}+\frac {3 i}{2}, 3, 2 i \left (x -4\right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-i x} \operatorname {KummerU}\left (\frac {3}{2}+\frac {3 i}{2}, 3, 2 i x -8 i\right ) \left (x -4\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{i x} y}{\left (x -4\right )^{2} \operatorname {KummerU}\left (\frac {3}{2}+\frac {3 i}{2}, 3, 2 i \left (x -4\right )\right )}\right ] \\ \end{align*}