Link to actual problem [1201] \[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (-x^{2}-6 x +1\right ) y^{\prime }+\left (x^{2}+6 x +1\right ) y=0} \] 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 {\operatorname {HeunC}\left (1, 0, -5, \frac {3}{2}, 9, -x \right ) {\mathrm e}^{-x} x}{\left (1+x \right )^{5}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} \left (1+x \right )^{5} y}{\operatorname {HeunC}\left (1, 0, -5, \frac {3}{2}, 9, -x \right ) x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-x} \operatorname {HeunC}\left (1, 0, -5, \frac {3}{2}, 9, -x \right ) x \left (\int \frac {\left (1+x \right )^{4} {\mathrm e}^{x}}{\operatorname {HeunC}\left (1, 0, -5, \frac {3}{2}, 9, -x \right )^{2} x}d x \right )}{\left (1+x \right )^{5}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} \left (1+x \right )^{5} y}{\operatorname {HeunC}\left (1, 0, -5, \frac {3}{2}, 9, -x \right ) x \left (\int \frac {\left (1+x \right )^{4} {\mathrm e}^{x}}{\operatorname {HeunC}\left (1, 0, -5, \frac {3}{2}, 9, -x \right )^{2} x}d x \right )}\right ] \\ \end{align*}