Link to actual problem [13995] \[ \boxed {\left (-3 x^{3}+3 x^{2}\right ) y^{\prime \prime }-\left (5 x^{2}+4 x \right ) y^{\prime }+2 y=0} \] 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 {x^{2}}{\left (-1+x \right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{2} y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (-2+5 x \right ) x^{\frac {1}{3}}}{\left (-1+x \right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{2} y}{\left (-2+5 x \right ) x^{\frac {1}{3}}}\right ] \\ \end{align*}