Link to actual problem [6953] \[ \boxed {x^{2} y^{\prime \prime }-x \left (1+3 x \right ) y^{\prime }+\left (1-6 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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}^{-3 x} y}{x \left (9 x^{2}+12 x +2\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{9 x \left ({\mathrm e}^{3 x} \left (x^{2}+\frac {4}{3} x +\frac {2}{9}\right ) \operatorname {expIntegral}_{1}\left (3 x \right )-\frac {x}{3}-\frac {1}{3}\right )}\right ] \\ \end{align*}