Link to actual problem [6924] \[ \boxed {2 x y^{\prime \prime }+\left (-x +2\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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 {x}{2}} \left (2+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} y}{2+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2+x \right ) {\mathrm e}^{\frac {x}{2}} \operatorname {expIntegral}_{1}\left (\frac {x}{2}\right )-2\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (2+x \right ) {\mathrm e}^{\frac {x}{2}} \operatorname {expIntegral}_{1}\left (\frac {x}{2}\right )-2}\right ] \\ \end{align*}