Link to actual problem [1344] \[ \boxed {9 x^{2} \left (x +5\right ) y^{\prime \prime }+9 x \left (5+9 x \right ) y^{\prime }-\left (5-8 x \right ) y=0} \] 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 {\operatorname {hypergeom}\left (\left [\frac {11}{3}+\frac {2 \sqrt {34}}{3}, \frac {11}{3}-\frac {2 \sqrt {34}}{3}\right ], \left [\frac {1}{3}\right ], -\frac {x}{5}\right )}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{3}} y}{\operatorname {hypergeom}\left (\left [\frac {11}{3}+\frac {2 \sqrt {34}}{3}, \frac {11}{3}-\frac {2 \sqrt {34}}{3}\right ], \left [\frac {1}{3}\right ], -\frac {x}{5}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {13}{3}+\frac {2 \sqrt {34}}{3}, \frac {13}{3}-\frac {2 \sqrt {34}}{3}\right ], \left [\frac {5}{3}\right ], -\frac {x}{5}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {13}{3}+\frac {2 \sqrt {34}}{3}, \frac {13}{3}-\frac {2 \sqrt {34}}{3}\right ], \left [\frac {5}{3}\right ], -\frac {x}{5}\right )}\right ] \\ \end{align*}