Link to actual problem [6472] \[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (5 x +4\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = -1\).
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 &= \frac {\sqrt {1+x}\, \left (x +4\right )}{\left (-1+x \right )^{\frac {7}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{\frac {7}{2}} y}{\sqrt {1+x}\, \left (x +4\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 \left (x^{2}+5 x +4\right ) \ln \left (x +\sqrt {x^{2}-1}\right )-11 \sqrt {x^{2}-1}\, \left (x +\frac {19}{11}\right )}{\sqrt {x^{2}-1}\, \left (-1+x \right )^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x^{2}-1}\, \left (-1+x \right )^{3} y}{3 \left (x^{2}+5 x +4\right ) \ln \left (x +\sqrt {x^{2}-1}\right )-11 \sqrt {x^{2}-1}\, \left (x +\frac {19}{11}\right )}\right ] \\ \end{align*}