Link to actual problem [2411] \[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x^{2}-\left (3 x^{2}+2\right ) y=0} \] 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 {{\mathrm e}^{x} \left (2 x -1\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-x} 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}^{-3 x} \left (2 x +1\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{3 x} y}{2 x +1}\right ] \\ \end{align*}