Link to actual problem [2407] \[ \boxed {x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }-5 y^{\prime } x +9 y=0} \] 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 {x^{3}}{\left (x^{2}-1\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{\frac {3}{2}} y}{x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{3}}{x^{2}-1}+\frac {x^{3} \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )}{\left (x^{2}-1\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{\frac {5}{2}} y}{x^{3} \left (\left (x^{2}-1\right )^{\frac {3}{2}}+\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) x^{2}-\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )\right )}\right ] \\ \end{align*}