Link to actual problem [2984] \[ \boxed {x^{2} y^{\prime \prime }+\frac {3 y^{\prime } x}{2}-\frac {y \left (x +1\right )}{2}=0} \] 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 {{\mathrm e}^{\sqrt {x}\, \sqrt {2}} \left (2 x -1\right )}{x \left (\sqrt {x}\, \sqrt {2}+1\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-\sqrt {x}\, \sqrt {2}} \left (\sqrt {x}\, \sqrt {2}+1\right ) y}{2 x -1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\sqrt {x}\, \sqrt {2}} \sqrt {\frac {2 \sqrt {2}\, x^{\frac {3}{2}}-\sqrt {x}\, \sqrt {2}+2 x -1}{\sqrt {x}\, \sqrt {2}-1}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{\sqrt {x}\, \sqrt {2}} y}{\sqrt {\frac {\left (2 x -1\right ) \left (\sqrt {x}\, \sqrt {2}+1\right )}{\sqrt {x}\, \sqrt {2}-1}}}\right ] \\ \end{align*}