Link to actual problem [6056] \[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x +y \left (1+x \right )=0} \] 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 {\operatorname {BesselJ}\left (0, 2 \sqrt {x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {BesselJ}\left (0, 2 \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {BesselY}\left (0, 2 \sqrt {x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {BesselY}\left (0, 2 \sqrt {x}\right )}\right ] \\ \end{align*}