Link to actual problem [5469] \[ \boxed {x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = \infty \).
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 &= {\mathrm e}^{\frac {1}{x}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {1}{x}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {1}{x}} \operatorname {Ei}_{1}\left (\frac {1}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{x}} y}{\operatorname {Ei}_{1}\left (\frac {1}{x}\right )}\right ] \\ \end{align*}