Link to actual problem [7026] \[ \boxed {x \left (1-2 x \right ) y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+18 y=0} \] 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 {7 x -2}{\left (2 x -1\right )^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{4} y}{7 x -2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{5} \left (40 x^{2}-56 x +21\right )}{\left (2 x -1\right )^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{4} y}{x^{5} \left (40 x^{2}-56 x +21\right )}\right ] \\ \end{align*}