Link to actual problem [1386] \[ \boxed {4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 (x^{2}+4\right )^{\frac {5}{8}}}{x^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{4}} y}{\left (x^{2}+4\right )^{\frac {5}{8}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {-\frac {1024}{25}+\left (x^{2}+4\right )^{\frac {5}{8}} \left (x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {13}{8}\right ], \left [2, 2\right ], -\frac {x^{2}}{4}\right )-\frac {32 \gamma }{5}+\frac {64 \ln \left (2\right )}{5}-\frac {64 \ln \left (x \right )}{5}-\frac {32 \Psi \left (\frac {5}{8}\right )}{5}\right ) 2^{\frac {3}{4}}}{x^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{4}} y}{-\frac {1024}{25}+\left (x^{2}+4\right )^{\frac {5}{8}} \left (x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {13}{8}\right ], \left [2, 2\right ], -\frac {x^{2}}{4}\right )-\frac {32 \gamma }{5}+\frac {64 \ln \left (2\right )}{5}-\frac {64 \ln \left (x \right )}{5}-\frac {32 \Psi \left (\frac {5}{8}\right )}{5}\right ) 2^{\frac {3}{4}}}\right ] \\ \end{align*}