Link to actual problem [2392] \[ \boxed {4 x^{2} \left (1-x \right ) y^{\prime \prime }+3 x \left (1+2 x \right ) y^{\prime }-3 y=0} \] 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 {\left (-1+x \right )^{\frac {13}{4}}}{x^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{4}} y}{\left (-1+x \right )^{\frac {13}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \left (-1+x \right )^{\frac {13}{4}} \operatorname {hypergeom}\left (\left [\frac {7}{4}, \frac {17}{4}\right ], \left [\frac {11}{4}\right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \left (-1+x \right )^{\frac {13}{4}} \operatorname {hypergeom}\left (\left [\frac {7}{4}, \frac {17}{4}\right ], \left [\frac {11}{4}\right ], x\right )}\right ] \\ \end{align*}