Link to actual problem [2941] \[ \boxed {x^{2} y^{\prime \prime }-\left (-1+2 \sqrt {5}\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 &= x^{-\frac {1}{2}+\sqrt {5}} \sinh \left (\sqrt {3}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, x^{-\sqrt {5}} y}{\sinh \left (\sqrt {3}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {1}{2}+\sqrt {5}} \cosh \left (\sqrt {3}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, x^{-\sqrt {5}} y}{\cosh \left (\sqrt {3}\, x \right )}\right ] \\ \end{align*}