Link to actual problem [2414] \[ \boxed {x^{2} y^{\prime \prime }+x \left (x -7\right ) y^{\prime }+\left (x +12\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is integer"}
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}^{x} y}{x^{6} \left (x^{2}-12 x +30\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{x^{2} \left (x^{4} \left (x^{2}-12 x +30\right ) \operatorname {expIntegral}_{1}\left (-x \right )+\left (x^{5}-11 x^{4}+20 x^{3}+12 x^{2}+12 x +12\right ) {\mathrm e}^{x}\right )}\right ] \\ \end{align*}