Link to actual problem [1427] \[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-3 y \left (x +3\right )=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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 {y}{x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-2-x^{2}+x +\frac {6}{x}-\frac {24}{x^{2}}+\frac {120}{x^{3}}\right ) {\mathrm e}^{-x}+x^{3} \operatorname {expIntegral}_{1}\left (x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} x^{3} y}{x^{6} \operatorname {expIntegral}_{1}\left (x \right ) {\mathrm e}^{x}-x^{5}+x^{4}-2 x^{3}+6 x^{2}-24 x +120}\right ] \\ \end{align*}