Link to actual problem [6921] \[ \boxed {2 x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (1+7 x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference 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 {x}}{\left (-1+x \right )^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{3} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x \left (3 x^{2}-10 x +15\right )}{\left (-1+x \right )^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{3} y}{x \left (3 x^{2}-10 x +15\right )}\right ] \\ \end{align*}