Link to actual problem [6963] \[ \boxed {x^{2} y^{\prime \prime }+3 x \left (1+x \right ) y^{\prime }+\left (1-3 x \right ) y=0} \] 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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {9 x}{2}+6+\frac {1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 x y}{9 x^{2}+12 x +2}\right ] \\ \end{align*}