Link to actual problem [13999] \[ \boxed {\left (9 x^{3}+9 x^{2}\right ) y^{\prime \prime }+\left (27 x^{2}+9 x \right ) y^{\prime }+\left (8 x -1\right ) y=0} \] 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 {1}{\left (1+x \right ) x^{{1}/{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (1+x \right ) x^{{1}/{3}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{{1}/{3}}}{1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right ) y}{x^{{1}/{3}}}\right ] \\ \end{align*}