Link to actual problem [2406] \[ \boxed {x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+4 y=0} \] 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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-2 x} y}{x^{2} \left (2 x +1\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= -\frac {y}{2 \left (-\frac {1}{2}+{\mathrm e}^{2 x} \left (x +\frac {1}{2}\right ) \operatorname {expIntegral}_{1}\left (2 x \right )\right ) x^{2}}\right ] \\ \end{align*}