Link to actual problem [6464] \[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y=0} \] 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 &= \frac {\left (2 \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}+3 \sin \left (\frac {\sqrt {3}\, x}{2}\right ) x \right ) {\mathrm e}^{\frac {x}{2}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{2 \,{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}+3 \,{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {\left (-3 \cos \left (\frac {\sqrt {3}\, x}{2}\right ) x +2 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) {\mathrm e}^{\frac {x}{2}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-\frac {x}{2}} y}{3 \cos \left (\frac {\sqrt {3}\, x}{2}\right ) x -2 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )}\right ] \\ \end{align*}