Link to actual problem [2963] \[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x^{2}-\left (2+3 x \right ) y=0} \] 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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} y}{x^{2} \left (x +4\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{3}+4 x^{2}\right ) {\mathrm e}^{x} \operatorname {expIntegral}_{1}\left (x \right )+2-x^{2}-3 x -\frac {2}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{{\mathrm e}^{x} x^{3} \left (x +4\right ) \operatorname {expIntegral}_{1}\left (x \right )-x^{3}-3 x^{2}+2 x -2}\right ] \\ \end{align*}