Link to actual problem [2412] \[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-9 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 {\left (-1+x \right )^{3}}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{\left (-1+x \right )^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (60-\frac {180}{x}+\frac {180}{x^{2}}-\frac {60}{x^{3}}\right ) \ln \left (-1+x \right )-63+3 x^{2}+15 x +\frac {9}{x}+\frac {81}{x^{2}}-\frac {47}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{60 \left (-1+x \right )^{3} \ln \left (-1+x \right )+3 x^{5}+15 x^{4}-63 x^{3}+9 x^{2}+81 x -47}\right ] \\ \end{align*}